
/ST3 




MATHEMATICS SIMPLIFIED 



AND 



MADE ATTRACTIVE; 



OR THE 



LAWS OF MOTION 



EXPLAINED. 



BY THOMAS FISHER. 



PHILADELPHIA: 

PUBLISHED DY THE AUTHOR, No. 110 80UTn FRONT STREET. 

1853. 



.0 . , 



a%' 






Entered according to Act of Congress, in the year 1853, by 

THOMAS FISHER, 

In the Clerk's Office of the District Court, for the Eastern District of 
Pennsylvania. 



Printed by JOHN C. ROBB, No. 8 Pom Street, Philadelphia. 



GREAT INDUSTRIAL EXHIBITION, 

HYDE PARK, LONDON, 1851. 

AWARD OF THE JURY 

PRESENTED TO THE ROYAL COMMISSION, 

INTS, AND Vti 
THEIR USE. 



JURY. 
Sir David Brewster, F. R. S., Chairman, Principal of the University of St. 

Andrews. 
Sir John Herschel, Bart., F. R. S. 
Richard Potter, A. M. University College, London. 
L. Mathieu, France, Member of Bureau of Longitude, &c. 
Baron Armand Seguier, France, Member of Institute, &c. 
Professor Colladon, Switzerland. 
Professor Schubarth, Prussia. 
Professor Hetsch, Denmark, 

E. R. Leslie, United States, Artist, Member of the Royal Academy. 
E. B. Dennison, 42 Queen Ann Street. 

W. H. Miller, F. R. S. Professor of Mineralogy, Cambridge. 
J. Glaisher, F. R. S. Observer in Greenwich Observatory. 
L. A. J. Quetelet, Belgium, Secretary to the Royal Academy at Brussels. 
Rev. W. S. Kingsley, Sidney College, Cambridge. 
Lord Wrottesley, 
J. S. Bowerbank, 3 Highbury Grove. 

Fisher, (United Slates, No. 263, p. 1452,) has exhibited " Mathematics Sim. 
plified," consisting of some beautifully-drawn diagrams, intended to facilitate the 
study of Mathematics. His idea is that of teaching a physical geometry, cither 
preliminary to, or when no better may be had, instead of the science. His 

iii 



lv AWARD OF THE JURY OF THE GREAT EXHIBITION. 

method of using the diagrams is by teaching each step by a course of reasoning, 
and illustrating the laws by well drawn figures. 

Nobody can question the great disadvantage under which students lie who 
have to apply geometry graphically, if their previous figures have been drawn 
only by hand ; or what is worse, if badly drawn by ruler and compasses. It is 
doubtful which of the two is the greater evil — the giving a student ruler and 
compass as part of his course of geometry, or the making reasoning on badly 
drawn figures the only preparation for a draughtsman or architect, &c. This 
has been often said, but seldom accompanied with any proof of the satisfactory 
use which may be made of well drawn diagrams. Mr. Fisher has the merit of 
offering this proof to the Exhibition, with some ingenious ideas as to the manner 
in which the details may be managed. The attention he has paid to one point, 
viz : the exhibition of areas of given simple ratios under the same and different 
forms, is particularly beneficial. Some of his diagrams are, in this respect, 
excellent studies for an eye which is to be trained to correct estimation. His 
method is particularly applicable, for adults possessed of a power of thought, 
which requires to be enlisted and exercised to make their study agreeable, or 
even profitable. 

Mr. Fisher's merit may be described as consisting in — 1st, the application of 
the idea of teaching by physical perception, to a wider range of subject than merely 
making very exact drawings of the propositions to be demonstrated in Euclid : 
2d, the ingenuity of his details ; 3d, the beauty of the drawings. 

The Jury considered Mr. Fisher deserving Honorable Mention 



CONTENTS. 



Award of the Jury. (Industrial Exhibition, London, 1851.) - - 3 

Preface, 9 

Introduction, 13 

CHAPTER I. 

On the Usefulness and Importance of Mathematics. 

Quotations from Locke, Swift, Bacon, Stewart and others, ... 23 

CHAPTER II. 

Definitions of Mathematics. 

Chambers, Johnson, Eulcr, Comte, 29 

Author's own Definition, 32 

OfArithmetic 33 

Author's Definition, 34 

Of Geometry, - 35 

Plato's criticism, 36 

Aristotle. The relation of Geometry to Arithmetic, ... - 37 

Author's Definition, 37 

OfAlgebra, 40 

Johnson, Newton, Chambers, Stewart, 41 

Author's Definition, * 43 

CHAPTER III. 

On the Nature of Demonstrative Evidence. 

Want of improved methods of instruction, 46 

Quotations from Dr. Beddoes and Prof. Gilbert, 50 

From Locke, 53 

CHAPTER IV. 

Explanation of the Plates. 

The three great divisions of measurement, 55 

PLATE No. 1. 

Measurement — Linear, Superficial, Massive, 56 

The relation of Elementary Geometry to the Laws of Motion, - - .58 

v 



VI CONTENTS. 

PLATE No. 2. 

The Parallelogram and the Triangle, and their relation to each other, - 59 

PLATE No. 3. 
Length and Breadth, and their relation to Fluent Quantities. Primary 

idea of Fluxions, Analysis of Infinities, 61 

PLATE No. 4. 

Aggregates of Triangles. The Triangle, Square, Pentagon, Hexagon, 

Octagon, and their approach to the Polygon, ... - . 64 

PLATE No. 5. 

The relation of the Polygon to the circle, that is, " squaring the circle," - 66 
The Method of Exhaustions of Archimedes. There is or can be none other. 67 
Squaring of the circle unmystified, and set at rest for ever, or a plain, 
intelligible, primary, and popular explanation of the Integral and 
Differential Calculus, - -69 

PLATE No. 6. 
Adjunct and subordinate to Plate No. 5, further illustrating differential 

approximation, 71 

PLATE No. 7. 
The measurement of all plane figures, whether bounded by straight or 

curve lines, is simply the result of Length and Breadth, 72 

PLATE No. 8. 

Description of the Plate, - 73 

Application of Geometry to Mechanics, 75 

Definition of Pure Mathematics, 76 

Explanation of the Lever, in all its varieties, 78 

The Wheel and Axle, the Pulley, the Balance, the Steelyard, - - - 79 

Natural Philosophy — Light — Perspective, 81 

Parallax — Celestial perspective, -------- 82 

Heat, Electricity, Sound, Gravitation. All radiant phenomena illustrate 

each other, 82 

PLATE No. 9. 

Same subjects continued, 83 

Familiar illustration of Light and Heat, (" squares of the distances,") - 84 



Gravity, exemplification of 86 

Laws of forces, to be observed in their results, Motion, &c. ... 86 

Centripetal and Centrifugal forces, 87 

Reference to Plate No. 5, 88 

Application to the orbits of the Planets and Comets, 89 

Kepler. Retrospect of the previous progress of Astronomy. Pythagoras. 

Copernicus. 90 

TychoBrahe, 91 

Hipparchus. Tycho. Bradley. --,,---- 92 

Kepler. Tycho. Galileo 93 

Kepler. The ellipticity of the orbit of Mars. His first law. - - 94 
The Second Law of Kepler. The Radius Vector. Equal areas in equal 

times, 95 

The Pendulum. Galileo, 97 

Notice of Galileo's researches, 98 

The Third Law of Kepler, 101 

Recapitulation of the Laws of Kepler, 102 

Sir John HerschcPs eulogy of their surpassing importance, - - - 104 

PLATE No. 10. 

Confirmation of the Laws of Kepler, by Cassini & Newton, - - - 105 
Their application by Cassini to the orbits of the satellites of Jupiter and 

Saturn, 105 

By Newton to the orbit of the great Comet of 1680, - - - - 106 
Retrospect of Newton's earlier trains of thought in 1665-6, in relation to 

the gravitation of the Moon, 107 

Picard, in 1679, measures an arc of the meridian, and thus afforded Newton 

correct data for determining the earth's radius, and enabled him to 

apply the laws of Kepler to the earth's satellite, the Moon, - - 107 

The comparative reciprocal attraction of the Planets, - - - - 108 
Their comparative mutual disturbance of each other's motions in their 

orbits determines their comparative weight, 109 

The discovery of the new planet Neptune, 109 

Mewton's generalization of universal gravitation, 110 

Its application to the irregularities of the Moon's orbit, - » - - 111 

Universal gravitation is but the generalization of the Laws of Kepler, - 112 

PLATE No. 11. 
A systematic series of diagrams, demonstrating the universality of tho 

problem of Pythagoras, 112 



VU1 CONTENTS. 

PLATE No. 12 
Further exemplification of the same problem, 115 

PLATE No. 13. 

Further exemplification of the same problem, which is capable of infinite 

variety, 115 

Pythagoras, 116 

PLATES No. 14 & 15. 
Further exposition of the same fundamental problem, - - - - 117 
Logarithms, 118 

PLATE No. 16. 
Further exemplification of the same problem, with an Algebraic expression 

which determines the Arithmetical relations of the series, - - - 118 

PLATE No. 17. 

The same series, with the addition of circles, beautifully exemplifying the 
application of this problem of Pythagoras to the Geometrical, Arith- 
metical, and Algebraic determination of rotary and radiant forces. 
The composition of two forces, geometrically exhibited, is but Length 
and Breadth. The ratio of diffusion of radiant phenomena is repre- 
sented by Length, Breadth, and Thickness. It is the relation of the 
Cube, or of the sphere to its radius. - - . - . - - 121 

Pythagoras, Archimides, Kepler, 121 

PLATE No. 18. 
Subordinate to Plate No. 17, in amplification of the same truths, - - 121 

PLATE No. 19. 
In further amplification of the same truths, which are capable of unlimited 

application, 122 

Poetical illustration of radiant phenomena. The Creation of Light, - 123 
Concluding Remarks, 128 



PREFACE. 

It must be acknowledged that by the existing methods, the 
science of Mathematics is seldom effectively and compre- 
hensively taught. Much time is wasted on it with compara- 
tively little good result to the great majority of students. It 
is the purpose of the present work to show that very great 
improvements can be made in the methods of instruction. 

It may be thought that Mathematics is the last subject in 
which important improvement is to be expected, and that even 
an apology needs to be offered for an attempt so novel and 
audacious. In the opinion of many persons, the very cob- 
webs which have gathered around this science have acquired 
a time-honored and venerable antiquity, and he must be a bold 
man who dares to stir them. 

If the history of the past affords a presage of the present, 
a storm of opposition may be expected to any great improve- 
ment in this subject, similar to that which, in Italy, assailed 
Galileo's revelation of the Solar System, and in England, 
during many years, suppressed Harvey's demonstration of the 
circulation of the blood. 

On the other hand, it is certain that the existing methods of 
teaching fail to interest the great majority of minds ; that 
there are at this moment living, but few mathematicians in 
any comprehensive sense. However true, however ingenious, 
however worthy of immortal honor, the problems which the 
great Greek Geometers have bequeathed to us, as presented 

b ix 



PREFACE. 



to the student in the pages of Euclid, these pages are to the 
great majority, emphatically tedious and repulsive ; they are 
very imperfectly comprehended, and therefore may practically 
be considered as little better than a sealed book; they are 
wanting in that simple, harmonious, and beautiful connection 
with each other, and with the glorious all-comprehensive 
superstructure of the physical laws, which would give them 
their highest value, and which has been reserved for a later 
age — an age subsequent to the labors of Galileo, of Tycho 
Brahe, and of Kepler. 

Heretofore the appeal has been made so entirely to the 
memory rather than to the understanding, that we are not to 
be surprised at the repulsiveness of the methods by which the 
world has been so long mystified. 

Mathematics, occupying as it does, a sort of intermediate 
position between physical science and mental philosophy, in- 
terwoven on the one hand with the forces and laws of Nature 
as well as of the Mechanic Arts, while, on the other hand as 
the science of absolute certainty, it is acknowledged to be the 
best means of cultivating a comprehensive and logical use of 
the intellectual faculties, is of indispensable importance to men 
of all professions ; to the scholar and to the man of science, 
to the logician and to the mechanic, to the artist and to the 
engineer, as well as to the naturalist and to the metaphysician. 

In Mathematics, the human intellect is justly considered as 
having gradually accomplished its most perfect achievement, 
revealing the noblest evidences of the wisdom, omnipotence 
and omnipresence of the Creator. The highways to its 



PREFACE. XI 

grandest results have been discovered and investigated by 
men of the greatest genius — those noble martyrs of science, 
who, often amid the neglect or persecution of their cotempo- 
raries, have passed their lives in patient devotion to their 
favorite study, have gloriously surmounted its rugged ac- 
clivities, and thus have entitled themselves to immortal grati- 
tude and honor. 

Though the compilation of Euclid was well for the age in 
which he lived, though it was a good text book twenty cen- 
turies ago, it may now fairly be assumed that the progress of 
the human mind, so conspicuous in every other department of 
industrial pursuits, and of physical science, imperatively de- 
mands for the nineteenth century a more efficient development 
of this great subject. 

The truths of Mathematics, the laws of Nature, have 
existed unchanged. Two hundred generations of men have 
lived and died, ignorant of these laws, to an extent not credit- 
able to the intellectual activity of our race. 

Any well sustained attempt therefore to improve this 
science, and to facilitate and insure its general acquisition, 
deserves attention. It is honorable, even to attempt to sim- 
plify, to explain and to make attractive, a subject which 
ought to be universally understood and universally popular. 

The art of Printing is certainly capable of achieving, 
and the demands of the age imperiously require a much 
better, as well as a much wider education among the teachers 
of the people and the people themselves. 



INTRODUCTION. 

Mathematics is a science in which nothing can be taken 
upon authority. It is emphatically a science of demonstra- 
tion, where self evident conviction alone, in the mind of each 
and every individual, is relied upon to command and to compel 
involuntary assent. 

What is demonstration? A demonstration 'is a chain of 
axioms. An axiom is a self evident or admitted truth, a 
stubborn fact that cannot be disputed. Self evident truths 
are simple results of observation, experiment and experience. 
Thus the apparently most difficult demonstrations rest upon 
the simplest truths. 

" To Demonstrate, verb active, [demonstro Latin,] to prove 
with the highest degree of certainty ; to prove in such a 
manner as reduces the contrary position to evident absurdity. 

"Demonstration, noun substantive, [demonstratio, Latin,] 
the highest degree of deducible or argumental evidence ; the 
strongest degree of proof ; such proof as not only evinces the 
position proved to be true, but shows the contrary to be ab- 
surd and impossible." Johnson's Folio Die. 1755. 

Words are the signs of ideas. Things, on the contrary, 
are the sources of ideas. An idea is a mental image ; an im- 
pression formed in the mind. An idea is an abstraction, when 
the source of it is not present. Thus, the idea or recollection 
of something that is past, is an abstraction. Thus, the general 
idea of justice, of truth, of natural philosophy, of mathematics, 
formed upon larger experience and education, is abstraction ; 
it is generalization. But an abstraction not founded upon, and 
not consonant with Nature and Truth, would be a falsity, an 
insanity. 

13 



14 INTRODUCTION. 

The child in his earliest progress in education, has every 
thing to learn by observation and experience. He learns not 
only that one finger and one finger make two fingers, but that 
of any one thing added to another similar thing, (one unit 
added to another unit) make two, &c. He thus has begun to 
acquire abstract ideas. An idea is a likeness. "Abstract" 
means, taken from. Every abstract idea must have been pre- 
ceded by a reality submitted to his senses, and more especially 
to his sight. An abstraction is a likeness. A likeness of 
what 1 Of something you have previously seen or known. 

Yon Humboldt has said that no two persons assign pre. 
cisely the same meaning to a word. The significance which 
is attached by each individual to any conventional language, 
is necessarily a result of his previous knowledge or education. 
All conventional language has for its base a natural language, 
a language of signs, a pantomimic language. 

The meaning of the simplest word which a child learns, is 
a result of observation or of education. The Abbe Sicard, 
in his researches into the origin of language in connection 
with the means of teaching the deaf and dumb, has, I think, 
established this fact. The word horse must be associated 
with the actual animal ; the word kindness must be associated 
with the natural expression of the emotion, before it can con- 
vey any meaning to the mind. The child first learns to dis- 
criminate a horse from a cow, and to associate the conven- 
tional words ; he very gradually becomes, by experience, ac- 
quainted with a considerable number of animals, and thus is 
capable of making more general comparisons ; in time he be- 
gins to observe their anatomy and comparative anatomy, and 
becomes an acute and philosophical observer. The idea of 
general classification at length occurs to the youthful Cuvier, 
and finally, in mature age, he writes his immortal work on 
the animal kingdom. 






INTRODUCTION. 15 

In like manner, the very simplest ideas of number, even 
that one and one make two, and that two and two make four, 
have to be acquired. A child, or an untaught savage may 
have a correct idea of numbers as far as he can count his 
fingers, but not beyond. It requires considerable familiarity 
with numbers to have an idea of hundreds, of thousands, or 
of millions, though we speak of them easily and thoughtlessly. 
Thus, the velocity of light, twelve millions of miles in one 
minute of time, though the fact that it is so, can be satisfacto- 
rily demonstrated, is too vast to be understood by uneducated 
persons without previous appropriate training. 

In a science which professes to be so clear and certain as 
Mathematics, it is almost incredible that the use of terms 
should be so vague and unintelligible as we find it at this ad- 
vanced period of the nineteenth century. There has evidently 
been too much assumption of profound meaning and too little 
honest intention to make intelligible to those of plain intellect 
and of common advantages, the truths of this noble and 
delightful science. 

It is to this cause probably that a very large part of the 
difficulties of its acquirement is to be attributed, and that the 
nomenclature of the most certain of sciences is so little un- 
derstood in an age like ours, boastfully claiming for itself an 
advanced intellectual and scientific culture. 

Lord Brougham has lately said in the English Parliament, 
" that there is not a schoolmaster in England who has not him- 
self need to go to school ;" and though the remark may not be 
very flattering to the professors of the age, it is probably true 
in a much wider and more emphatic sense even than its 
author intended. 

In this country we have a class of public men who make it 
their ambition to be considered friends of education, and who 
are always ready to flatter us with the idea that our system 



16 INTRODUCTION. 

of education is in a most flourishing condition. But with all 
this boasting it may be gravely doubted whether, relatively to 
the population, the number of men of really sound and compre- 
hensive education is equal to what it was half a century ago. 

Lord Bacon, in his much celebrated work, the "Novum 
Organon, or Rules for conducting the Understanding in 
search of Truth," published in 1620, Aphorism 90, says : 

" In the customs and institutions of schools, universities, 
colleges, and the like conventions, destined for the seats of 
learned men, and the promotion of knowledge, all things are 
found opposite to the advancement of the sciences ; for the 
readings and exercises are here so managed, that it cannot 
easily come into any one's mind to think of things out of the 
common road. Or if here and there one should venture to 
use a liberty of judging, he can only impose' the task upon 
himself, without obtaining assistance from his fellows ; and if 
he could dispense with this, he will still find his industry and 
resolution a great hindrance to the raising of his fortune. 
For the studies of men in such places are confined and pinned 
down to the writings of certain authors, from which, if any 
man happens to differ, he is presently reprehended as a dis- 
turber and innovator. But there is surely a great difference 
between arts and civil affairs ; for the danger is not the same 
from new light, as from new commotions. In civil affairs; it 
is true, a change even for the better is suspected, through fear 
of disturbance; because these affairs depend upon authority, 
consent, reputation and opinion, and not upon demonstration. 
But arts and sciences should be like mines, resounding on all 
sides with new works and farther progress. And thus it 
ought to be, according to right reason ; but the case, in fact, is 
quite otherwise ; for the above-mentioned administration and 
policy of schools and universities, generally opposes and 
greatly prevents the improvement of the sciences." 



INTRODUCTION. 17 . 

Baden Powell, the present distinguished Professor of Ma- 
thematics in the university of Oxford, in his " History of 
Natural Philosophy" published in 1842, quotes and endorses 
this aphorism in the following paragraph : 

" Bacon in proposing his new scheme of a philosophical in- 
stitution, is of necessity led to justify the proposal by a refer- 
ence to the services of existing institutions, and by showing 
how little they had done, or by their very nature and con- 
stitution were likely to do towards the real advancement of 
science. He censures, with severity indeed, but without bit- 
terness, in strong terms, but with a masterly exposition of the 
facts, which evinces the perfect justice of his condemnation, 
the system of the colleges and universities of his day. [Nov. 
Organon, 1 Aph. 90.] He observes that the lectures and ex- 
ercises were all of such a nature that no deviation from the 
established routine was likely to be thought of; that if a soli- 
tary attempt were made, the whole burden of it would rest on 
the individual, who would, moreover, find such attempts a 
serious impediment to his own advancement. The studies of 
these places were confined to a certain set of authors, as it 
were, imprisoned within those limits ; any one who should 
venture to deviate from this course, would be immediately 
condemned as a turbulent innovator. But, he adds, in the 
arts and sciences, as in mines, the whole region ought to re- 
sound with new works and further advances. Such was the 
state of the case in his day ; nor shall we find it much better 
in later times ; for the present it must suffice to remark that, 
in regard to Oxford, when the scholastic forms were in great 
measure broken up under the reign of Cromwell, the favorable 
opportunity which was afforded for establishing a better sys- 
tem in their place was, as we have seen, not lost by a few 
.ardent friends of true science ; owing however to a variety 
of causes, their partial attempts failed, as far as the university 

c 



18 INTRODUCTION. 

was concerned, and with the return of the Stuarts, the old 
system was re-established in all its authority." 

Jared Sparks, on assuming the Presidency of Harvard 
University, June 20, 1849, says : 

" Such have been the prodigious discoveries in the sciences, 
that the circle of what is called a liberal education has been 
greatly enlarged since the days of our fathers. The plan of 
instruction must be expanded in due proportions, and it is a 
problem difficult even in theory, but much more so in practice, 
how far one branch shall be curtailed to make way for 
another, which of them shall have the preference, and in what 
order the whole si. all be so adjusted that each may fall into 
its appropriate place, and contribute its appropriate share. If 
you appeal to the general voice, you will be met by a conflict 
of opinions that would multiply rather than clear away the 
difficulties. Each man would judge according to his own 
ideas of the relative importance of particular studies ; ideas 
derived rather from individual predilections, than from a com- 
prehensive view of the whole subject. Some value lightly 
every kind of study which does not bear the stamp of direct 
utility. Others would have the thinking and reasoning facul- 
ties mainly cultivated ; while others again would recommend, 
as the highest achievement, the art of communicating thoughts 
by a mastery over language, so as to captivate and convince 
a listening multitude, or impress great truths and brilliant 
conceptions on the reading world. # # # # 

" There is so general an opinion among students, particularly 
in their early college life, that the mysteries of the Mathema- 
tics are above their comprehension, the mathematical faculty 
being, as they suppose, a gift of nature conferred on a few 
favored mortals only, that it might be hazardous to deny a 
position maintained by such numbers, although it may be 
doubted whether it is the result of resolute and laborious ex- 



INTRODUCTION. 19 

periment on their part. To accommodate such misgiving 
minds, however, it may not be amiss to leave Mathematics, at 
the proper stage of the college course, in the category of 
elective studies." #*##** 

The distinguished and learned gentleman having thus briefly 
and coldly dismissed the study of mathematics, proceeds to 
pass a very strong, just, and eloquent encomium on the value 
of the study of the ancient classics. 

" For the last thousand years, at all times, and in all coun- 
tries, where learning of any sort has been thought worthy of 
attention, under every form of government, and at every stage 
of civilization, the ancient classics have made a prominent part 
in every system of education. It is a significant fact, also, 
that the numerous colleges in our extended republic it is 
believed, have followed this example. # # # 

" The highest attainment of a man of action and of thought 
is the art of using language with accuracy, elegance, force, 
and effect." ###### 

True ! But how is the man of action and the man of 
thought to originate 1 How are boys to be trained, so as to 
become men of thought ? is the question. 

It must be admitted that the mere art of using language is 
not necessarily indicative of valuable thought. And there is 
also another side to this question, as it is often possible 
to teach acknowledged idiots to talk ; as it is possible to 
teach persons of various grades of intellect above idiocy, 
to talk flippantly, fluently, and foolishly, so it is possible to 
teach men of higher grades of intellect, and even of naturally 
excellent abilities, to make talking in an ostentatious and 
pedantic style a habit and an ambition, and thus immensely 
to overrate themselves, and their true value to society. 

It ought to be taught, and generally acknowledged, that to 
listen well, to know when to listen, and when to speak, is one 



20 INTRODUCTION. 

of the highest accomplishments of a disciplined intellect. We 
want schools like that of Pythagoras, where students may 
go through a regular probationary course of silent listening 
and observation. 

" The first step toward being wise is to know that we are 
ignorant." But most of us have not yet made even this first 
advance. Let us quote Locke, the greatest of mental phi- 
losophers : 

" If we would speak of things as they are, we must allow 
that all the art of rhetoric, besides order and clearness, all the 
artificial and figurative application of words eloquence hath 
invented, are for nothing else but to insinuate wrong ideas, 
move the passions, and thereby mislead the judgment, and so, 
indeed, are a perfect cheat : and, therefore, however laudable 
or allowable oratory may render them in harangues, and 
popular addresses, they are certainly, in all discourses that 
pretend to inform and instruct, wholly to be avoided; and 
where truth and knowledge are concerned, cannot but be 
thought a great fault, either of the language, or person that 
makes use of them. What and how various they are, will be 
superfluous here to take notice : the books of rhetoric which 
abound in the world will instruct those who want to be 
informed ; only I cannot but observe how little the preserva- 
tion and improvement of truth and knowledge is the care and 
concern of mankind, since the arts of fallacy are endowed and 
preferred. 'Tis evident how much men love to deceive and 
be deceived, since rhetoric, that powerful instrument of error 
and deceit, has its established professors, is publicly taught, 
and has always been had in great reputation ; and I doubt not 
but it will be thought great boldness, if not brutality, in me to 
have said thus much against it." 

It is one of the evils of our age and country, some parts of 
it especially, for talkers, men whose chief ambition it is to 
hear themselves talk, to usurp a position to which their think- 



INTRODUCTION. 21 

ing capacities are utterly disproportioned. The business of 
our legislative halls, our legal tribunals, our town meetings, is 
often impeded by this rhetorical verbiage, to an extent which 
demands abatement. 

True eloquence is nature; it is the natural expression of 
thought, which art but attempts to counterfeit. 

A skilful use of language, a comprehensive knowledge of 
things, and a wise and reflective sagacity in discriminating 
the relations of causes and effects, so far from being antago- 
nistic or opposite, are in truth intimately subservient and 
essential to each other ; and as an accomplished education is 
composed of an aggregate of many acquirements, whatever 
really facilitates, and thus saves time in the acquisition of one 
study, undoubtedly promotes the whole. 

It is not only a false, but a very narrow and injurious 
doctrine, which supposes that a knowledge of those immortal 
discoveries of the Greek geometers, which form the basis of 
modern science, should excite in any degree less interest for 
those other forms in w T hich Grecian intellect has embodied 
itself, for the veneration of posterity. 

Would it not be absurd to suppose that a knowledge of the 
Greek language, which is undoubtedly the most brilliant 
achievement of human genius, and the most remarkable memo- 
rial of the earliest real civilization, should diminish the estima- 
tion of Greek sculpture or Greek architecture 1 yet this would 
not be more absurd than for this knowledge to diminish in the 
mind of one who understands them, the value of the geomet- 
rical discoveries of Pythagoras, " the wisest of the Greeks." 



CHAPTER I. 

ON THE USEFULNESS AND IMPORTANCE OF MATHEMATICS. 

The usefulness of Mathematics is twofold. The first con- 
sists in its universal relation to physical science. The second 
in its relation to mental philosophy. It is perhaps equally 
valuable in both respects, as essential to every liberal and 
useful system of education. 

It is important to men of every profession, as the means 
by which alone they are capable of understanding those 
principles which form the basis of all physical science — of all 
the physical laws of Nature. It may be said to form the 
framework of all the Sciences, and of all the Arts, or at least 
of all those of which Mechanics or Natural Philosophy 
constitute a part. 

It is also, as the science in which our perceptions, reason- 
ings, and conclusions are the clearest and least subject to 
doubt or dispute ; that in which our most important mental 
habits are formed and exercised, and in the prosecution of 
which steady and continued thought is indispensable, acknow- 
ledged to be the best means of cultivating a logical and com- 
prehensive use of our intellectual faculties. Mathematics 
ought to be taught faithfully, and then they would be learned 
with delight by all persons who are ambitious of a good and 
sound education. 

Instead, however, of extending his own views on the sub- 
ject, the author begs to offer the following quotations from 
acknowledged authorities : 

" Would you have a man reason well, you must use him to 
it betimes, exercise his mind in observing the connection of 
ideas, and follow them in train. Nothing does this better than 
mathematics, which, therefore, I think should be taught all 

23 



24 USEFULNESS AND IMPORTANCE 

those who have the time and opportunity, not so much to 
make them mathematicians, as to make them reasonable 
creatures." 

[Locke, on the Conduct of the Human Understanding, Intro- 
duction, Sec. 6. 

" It is to be observed that as there is no study which may 
be entered upon with a less stock of preparatory knowledge 
than mathematics, so there is none in which a greater number 
of uneducated men have raised themselves by their own 
exertions to distinction and eminence."' — [Dean Swift. 

" The mathematicians, from very plain and easy beginnings, 
by gentle degrees, and a continued chain of reasonings, pro- 
ceed to the discovery and demonstration of truths that appear 
at first sight beyond human capacity." * _ # # # 
[Locke on the Improvement of our Knowledge, Bk. 4, ch. 12, s. 7. 

" In mathematics, and in natural philosophy, since mathe- 
matics was applied to it, we see the noblest instances of the 
force of the human mind, and of the sublime heights to which 
it may rise by cultivation." 

[Dugald Stewart, vol. 3, chap. 1, sec. 2. 

" I have mentioned Mathematics as a way to settle in the 
mind a habit of reasoning closely, and in train; not that 
I think it necessary that all men should be deep mathema- 
ticians, but that having got the way of reasoning which that 
study necessarily brings the mind to, they might be able to 
transfer it to other parts of knowledge as they have occasion; 
for in all sorts of reasoning, every single argument should be 
managed as a mathematical demonstration, the connection 
and dependence of ideas should be followed, till the mind is 
brought to the source on which it bottoms, and observes the 
coherence all along." 
[Locke, Introduction to the conduct of the Human Under- 

standing, Bk. 4, Sec. 7. 



OF MATHEMATICS. 25 

" If a man's wit be wandering, let him study the mathe- 
matics ; for in demonstrations, if his wit be called away ever 
so little, he must begin again." — [Lord Bacon. 

" The habits of the mathematician afford exercise to that 
species of attention which enables us to follow long processes 
of reasoning, and to keep in view all the various steps of an 
investigation till we arrive at the conclusion. In mathematics 
such processes are much longer than in any other science, 
and hence the study of it is peculiarly calculated to strengthen 
the power of steady and concatenated thinking — a power 
which in all the pursuits of life, whether speculative or active, 
is one of the most valuable endowments we can possess." 

[Dugald Stewart, Part III, chap. 1, sec. III. 

The recognition of the importance of mathematics to medi- 
cine was cotemporaneous with the early labors of the founders 
of the science. 

In the fourth century before Christ, Hippocrates, " the 
Father of Medicine," himself a distinguished mathematician, 
commended its study to his son, that he might more perfectly 
understand the situations, dislocations, contacts, and action of 
the bones. 

In the next generation, Aristotle, amid various allusions to 
mathematics, particularises its connections with medicine. 

Dr. Samuel Jackson, one of the most distinguished of 
American physicians, in a lecture introductory to a course on 
the Institutes of Medicine, in the University of Pennsylvania, 
October 20, 1847, says: 

" The branches of education the best adapted to prepare 
the mind for the study of medicine, in its new relations and 
character, are geometry, or the higher mathematics, physics, 
and chemistry. 

" The objects of education should be to discipline the mind 
in the intellectual process by which it acquires knowledge ; 

D 



26 USEFULNESS AND IMPORTANCE 

and in the methods of investigating and analyzing, by rigid 
scrutiny, the complexities of the compound phenomena or 
facts of science, by which it lays bare the truth ; just as the 
brilliant gem is separated from the dross in which it is 
concealed. 

" The spirit of geometry pervades all the physical sciences, 
and rules over all their operations and experiments. From 
the clearness, precision and order it imparts to mental opera- 
tions, it may be looked upon as almost indispensable for the 
successful cultivation of science. 

" He who possesses it is enabled to discover the fine thread, 
invisible to others, that leads through the mazes of the laby- 
rinth that surrounds the shrine of truth and guides to its 
portals. He who has it not wastes his powers in fruitless 
efforts, or at best, meets with partial success in imperfect 
results. ######### 

" The animal economy of man in its diversified relations 
and conditions, physiological and pathological, is an exceed- 
ingly complicated mechanism. ##**#* 
A knowledge of physics and geometry, the ruling spirit of all 
scientific investigations, must lend the most material assist- 
ance in obtaining correct views of the structure of the animal 
economy, the nature of its operations, and the laws which 
preside over them. In this respect, this knowledge is of the 
greatest importance to the medical student. 
Extract from a letter of Vice Admiral, Lord Collingwood, one 
of the ablest of seamen and naval chieftains, written after 
the battle of Trafalgar, to Lady Collingwood, on the educa- 
tion of their daughters. 

Ocean, June 16, 1806. 
######## 

" How do the dear girls go on ? I would have them taught 
geometry, which is of all sciences in the world the most 



OP MATHEMATICS. 27 

entertaining : it expands the mind more to the knowledge of 
all things in nature, and better teaches to distinguish between 
truths, and such things as have the appearance of being 
truths, yet are not, than any other. ***** 
How would it enlarge their minds, if they could acquire a 
sufficient knowledge of mathematics and astronomy to give 
them an idea of the beauty and wonders of creation ! I am 
persuaded that the generality of people, and particularly fine 
ladies, only adore God because they are told it is proper, and 
the fashion to go to church ; but I would have my girls gain 
such knowledge of the works of the Creation, that they may 
have a fixed idea of the nature of that Being who could be the 
author of such a world." 



CHAPTER II. 

DEFINITIONS. 

Locke says, " 'tis ambition enough to be employed as an 
under labourer in clearing ground a little, and removing some 
of the rubbish that lies in the way to knowledge, which cer- 
tainly had been very much more advanced in the world, if the 
endeavours of ingenious and industrious men had not been 
much cumbered with the learned but frivolous use of uncouth, 
affected, or unintelligible terms, introduced into the Sciences, 
and there made an Art of, to that degree, that philosophy, 
which is nothing but the true knowledge of things, was thought 
unfit, or incapable to be brought into well bred company, 
and polite conversation. 

" Vague and insignificant forms of speech, and abuse of lan- 
guage, have so long passed for mysteries of science; and 
hard or misapplied words, with little or no meaning, have by 
prescription such a right to be mistaken for deep learning, 
and height of speculation, that it will not be easy to persuade 
either those who speak, or those who hear them, that they are 
but the covers of ignorance, and hindrance of true knowledge. 
To break in upon this sanctuary of vanity and ignorance, 
will be, I suppose, some service to the human understanding." 

In any dictionary or cyclopedia extant, with which I am 
acquainted, the word Mathematics has not as it seems to me, 
a good, popular definition. 

I have accordingly attempted to present one. But, in the 

first place, at the risk of being tedious, I have thought it just 

to collate all of the existing definitions within my reach, and 

of these, to transcribe several which have appeared to me 

28 



DEFINITIONS. 29 

most valuable or peculiar, avoiding many which are evidently 
mere copies. 

Mathematics, from the Greek noun fia&tfia [learning,] and 
the Greek verb pav&avw [to learn,] disco, intelligo, doceor, 
interpreto, sentio, imbuor, comperio, deprehendo, percipio, 
cognosco." — [Constantino 's Lexicon, 1592. 

"Mathematics, sciences or arts taught by demonstration, 
and comprehend four of the liberal sciences, Arithmetic, 
(wherein Algebra is comprehended) Geometry, Music, As- 
tronomy." 

[Glossographia, or Dictionary of Hard Words, London, St. 
Paul's Church Yard, 1656.] 

With the ancients, mathematics meant all sorts of learning 
and discipline. To Plato, founder of the Greek school of 
mathematics, is attributed the earliest distinct discrimination 
of physical, mental, and moral science. Mathema, in the 
Greek vocabulary, more accurately meant physical science in 
all its amplifications, &c. 

" Mathematics, the science of quantity ; or a science that 
considers magnitudes either as computable or measurable." 

" See Quantity and Magnitude." 

" This word in its original (iadi]aig signifies discipline or 
science in the general, and seems to have been applied to the 
doctrine of quantity, either by way of eminence, or by reason 
of this having the start of all other sciences, the rest took the 
common name therefrom. See Science. 

" Pure Mathematics consider quantity abstractedly, and 
without any relation to matter or bodies. 

"Mixed Mathematics consider quantity as subsisting in ma- 
terial being : for example, length in a road, breadth in a river, 
height in a star or the like. 

"Mixed Mathematics is Applied Mathematics. It is suf- 
ficient to determine an art to be a branch of mixed mathe- 



30 DEFINITIONS. 

matics that pure mathematics are applicable thereto, i. e. 
that it may be explained and demonstrated from the prin- 
ciples of arithmetic and geometry. Such are Mechanics, 
which consider motion, or the laws of moving bodies. 
Hydrostatics, which consider the laws of fluids, or of bodies 
gravitating in fluids. Pneumatics, the air, with regard to the 
laws and mensuration thereof. Hydraulics, the motion of 
fluids. Optics, direct light or vision. Catoptrics, reflected 
vision — Reflection. Dioptrics, refracted vision — Refraction. 
Perspective, the images of objects, in order to delineate or 
represent them. Astronomy, the Universe, and the phenomena 
of the heavens. Geography, the Earth, both as in itself and 
in its affections. Hydrography, the sea, principally as navi- 
gable. — Navigation. Chronology, Time, with regard to the 
measuring and distinguishing thereof. Gnomics, or dialling, 
shadows in order for determining the hour of the day. Mili- 
tary Architecture, fortification the strength of places with 
regard to their defence against an enemy. Civil Architecture, 
building &c." 

[Chamber 's Cyclopedia of Arts and Sciences, London, 1743. 

" Mathematics, n. s. iia&7][iazixa the science which contem- 
plates whatever is capable of being numbered or measured ; 
and it is either pure or mixed ; pure considers abstracted 
quantity, without any relation to matter ; mixed is interwoven 
with physical considerations." 

[Dr. Johnson's Folio Dictionary, London, 1755. 

" Mathematics, in general, is the science of quantity ; or, the 
science which investigates the means of measuring quantity. 

" Whatever is capable of increase or diminution, is called 
magnitude or quantity. 

" Quantity is what is made up of parts, or is capable of being 
greater or less." — [Maclaurin. 

" A sum of money, therefore, is a quantity, since we may 



DEFINITIONS. 31 

increase it and diminish it. It is the same with a weight, and 
other things of this nature. 

" From this definition it is evident that the different kinds of 
magnitude must be so various as to render it difficult to enu- 
merate them ; and that is the origin of the different branches 
of the Mathematics, each being employed on a particular kind 
of magnitude. 

"Now, we cannot measure or determine any quantity except 
by considering some other quantity of the same kind as 
known, and pointing out their mutual relation. If it were 
proposed, for example, to determine the quantity of a sum of 
money, we should take some known coin, as a louis, a crown, 
a ducat, a dollar, or some other coin, and show how many of 
these pieces were contained in the given sum. 

" In the same manner if it were proposed to determine the 
quantity of a weight, we should take a certain known weight, 
for example a pound, an ounce, and then show how many 
times one of these weights is contained in that which we are 
endeavoring to ascertain. If we wished to measure any length, 
or extension, we should make use of some known length, such 
as a foot." — [Leonard Euler, 1760. 

" Mathematics treat of magnitude and numbers, instructing 
us how to measure, estimate and compute the different dis- 
tances, magnitudes, and motions of bodies, with respect to one 
another." 

[Horne. State of the case between Newton and Hutchinson. 

"Mathematics, the science which investigates the conse- 
quences which are logically deducible from any given or 
admitted relations between magnitudes or numbers." 

[Brande's Encyclopedia of Science. 

" Mathematics the science of number and measure ; the 
science of quantity ; the science which treats of magnitude 
and number, or whatever can be numbered or measured." 

[JVoah Webster, Hartford, 1806. 



32 DEFINITIONS. 

" We are now able to define mathematical science with pre- 
cision, by assigning to it as its object the indirect measurement 
of magnitudes, and by saying it constantly proposes to deter- 
mine certain magnitudes from others by means of the precise 
relations existing between them" 

\Auguste Comte's Philosophy of Mathematics. 

Let us ask ourselves, what is mathematics? It is the 
science of measurement. It measures quantities and numbers, 
magnitudes and multitudes. It teaches us to measure and to 
count. 

Mathematics, the science of measurement, teaches us to 
measure accurately things of all sorts, of all sizes, of all forms. 
By mathematics we measure not only distances, lines, linear 
measurements, — surfaces, areas, superficial measurements, — 
spaces, masses, solid contents or cubical measurements ; but 
also forces, motions, velocities, intensities, times, weights, 
and the various relations of all and each of these, respectively 
to each other, and to the simplest forms of measurement. 

Mathematics uses numbers as one of its means of measur- 
ing. That part of mathematics which treats of numbers is 
called Arithmetic. Arithmetic is the use of numbers. Arith- 
metic is literally number measure. Algebra is a contrivance 
which records or represents concisely, mathematical truths. 
Algebra is in fact only a method of short hand. Algebra is 
to mathematics what stenography is to writing ; a short hand 
method of recording what is already known. 

Mathematics is a certain and unchangeable science. It is 
the most comprehensive means of interpreting the immutable 
laws of Nature. 

Mathematics begins with the simplest ideas of number and 
of measurement, which almost every child understands, and 
connects these simplest measurements like the links of a chain 
very naturally and beautifully with other measurements, which 



DEFINITIONS. 33 

at first sight appear difficult ; but if properly approached, the 
more advanced results can be reached with ease and pleasure, 
as a child by a graded pathway may delight to ascend a high 
mountain whose aspect from a distance was discouraging to 
the effort. 

Mathematics, beginning with the simplest measurements, 
includes all measurements. It determines all the measure- 
ments used in the workshop of the mechanic, in the market- 
place, and in the exchanges and commerce of nations. It 
comprises all adjustments of machines and mechanical calcu- 
lations used in contriving, estimating, and constructing rude 
building materials into the most skilfully wrought houses or 
ships. Its applications include all labor-saving contrivances, 
from the most delicate cotton spindle to the colossal steamship. 
It defines the principles of drawing and perspective. On the 
one hand it makes the minuter microscopic measurements, and 
investigates the structure, crystalization, and organization of 
minerals, plants, and animals, while on the other, by the aid of 
the telescope (and the genius of the immortal Kepler,) it 
connects by simple laws of boundless comprehensiveness 
all terrestrial perspective, — all terrestrial mechanics, with 
astronomy or celestial mechanics, by which are determined 
with wonderful accuracy the distances, the times of revolu- 
tion, the transits, the masses, and the comparative weights of 
the planets. 

Mathematics, the science of measurement, is in ordinary 
language divided into 

Arithmetic, — Number measure. 

Geometry,— Land measure. 

Algebra, — Letter or symbol calculation. 
" Arithmetic is a science which explains the properties of 
numbers, and shows the method of computing by them." 

[Encyclopedia Brittannica, 1th edition. 



34 DEFINITIONS. 

" Arithmetic, n. s. aQi&pog [number,] and [isrQt'co [to measure,] 
the science of numbers ; the art of computation. 

" We have very little intelligence about the origin and inven- 
tion of arithmetic ; but probably it must Uave taken its rise 
from the introduction of commerce, and consequently be of 
Tyrian invention. From Asia it passed into Egypt, where it 
was greatly cultivated. From thence it was transmitted to 
the Greeks, who conveyed it to the Romans, with additional 
improvements. But from some treatises of the ancients, .re- 
maining on this subject, it appears that their Arithmetic was 
much inferior to that of the moderns." 
[Quoted from Chamber's Encyclopedia into Dr. Johnson's 

Folio Dictionary. 

Mr. Peacock, in his very elaborate history of Arithmetic, 
contained in the Encyclopedia Metropolitana, shows that very 
simple and practical methods of numeration, founded upon the 
most obvious scales of numeration, the fingers and toes of the 
hands and feet, have among all nations preceded the formation 
of numerical language. 

In Arithmetic we have very many contrivances for simpli- 
fying and amplifying the use of numbers ; of which a single 
example may suffice. In numeration, the value of a number 
depends not only upon the simple number for which it stands 
when alone, but upon the place in which it stands. Thus, 
in 888, the three eights mean eight, eight tens, and eight 
hundreds. 

The place of a figure considered as affecting its value is 
determined by the column in which it stands, and in the 
absence of succeeding figures to indicate the existence of 
other columns, their place is supplied by cyphers, which of 
themselves are considered as having no value. Thus, the 
8 in 800 is of the same value as that in 863. 

The early Greeks used the square, as the square of eight, 



DEFINITIONS. 35 

the square of ten, the square of twelve, which term is bor- 
rowed from Geometry, and which well represents the multipli- 
cation table, as illustrated in the first Plate of this work, and 
which in fact is a very natural method of Arithmetic applied 
to Geometry. The invention of our common multiplication 
table is attributed to Pythagoras, who lived in the sixth cen- 
tury before Christ. 

Arithmetic has for its primary divisions Addition, Subtrac- 
tion, Multiplication, and Division. But Multiplication rightly 
defined is a wholesale addition. Division, in like manner may 
be said to be a short method of Subtraction. 

•' Multiplication rightly defined is a compendious addition." 
\W. Ludlam, St. John's College, Cambridge. 

The fundamental operations of arithmetic, as given in the 
Lilavati of the Hindoos, and as adopted in many of the books 
of arithmetic of the sixteenth century, are eight in number, viz : 
Addition, Subtraction, Multiplication, Division, the Square, 
Square root, Cube, Cube root. To the first four of these, the 
Arabs added two, viz : Duplation and Mediation, doubling and 
halving. It is quite possible, that, as a primary analysis of 
arithmetic, this is simpler, and more natural, and more easily 
understood than the very numerous rules of the moderns. 

" Geometry, n. s. yecoiiEtQia [geometric, French,] originally 
signifies the art of measuring the earth, or any distances or 
dimensions on or within it ; but it is now used for the science 
of quantity, extension, or magnitude, abstractedly considered, 
without any regard to matter." 

" Geometry very probably had its first rise in Egypt, where 
the Nile annually overflowing the country, and covering it 
with mud, obliged men to distinguish their lands one from 
another, by the consideration of their figure; and after which 
'tis probable, to be able also to measure the quantity of it, and 



36 



DEFINITIONS. 



to know how to plot it, and lay it out again in its just dimen- 
sions, figure and proportion ; after which it is likely a farther 
contemplation of those draughts and figures helped them to 
discover many excellent and wonderful properties belonging 
to them, which speculations were continually improving, and 
are still to this day* Geometry is usually divided into specu- 
lative and practical ; the former of which contemplates and 
treats of the properties of continued quantity abstractedly, 
and the latter applies these speculations and theorems to use 
and practice, and to the benefit and advantage of mankind." 
[Quoted from Harris into Dr. Johnson' s Folio Die. 1755. 

" In the muscles alone, there seems to be more Geometry 
than in all the artificial engines in the world." 

[Ray on the Creation. Dr. Johnson, 1755, 

" Geometry, originally and properly the art of measuring the 
earth, or any distances or dimensions on it. But Geometry 
now denotes the science of magnitude in general, compre- 
hending the doctrine and relations of whatever is susceptible 
of augmentation and diminution ; as the mensuration of lines, 
surfaces, solids, velocity, weight &c. with their various 
relations." — [Webster's Dictionary, 1828. 

Geometry, n. s. from ytj [the Earth,] and (teTQsoo [to mea- 
sure,] is the act of measuring the earth. This word ori- 
ginally meaning land measure, and embracing only a limited 
significance, had very improperly, as Plato thought, grown 
in his time into use as a substitute for the word [ihQtjgig 
which he very much preferred, and which meant measure- 
ment in its most comprehensive sense. 

The meaning generally attached to the word geometry, in 
its modern use, not unfrequently includes this comprehensive 
signification, originally and properly attached to the general 
word mathematics. 



DEFINITIONS. 37 

Thus it is said that the principles of mechanics and of 
astronomy, (that is, of terrestrial and celestial mechanics,) 
are, purely considered, geometrical considerations. That 
conic sections are geometrical considerations, and in fact all 
the relations of length, breadth, and thickness, and of the 
forces, motions, or results which these represent are often 
said to be geometrical truths. 

Thus as mathematics include multitude and magnitude — 
multitude as it is numerable, and magnitude as it is measur- 
able ; it may be defined to be the reciprocal comparison of 
multitude and magnitude. The modern use of the word 
geometry comprises all mathematics that is not arithmetic, 
and arithmetic is in truth only the application of numbers to 
measurement — number measure. " Measure," says Aristotle, 
" is that whereby quantity is known ; but quantity is known 
as quantity, either by unity or by number." No geometrical 
argument is of force which agrees not exactly with an arith- 
metical calculus ; and consequently all false reasonings in 
geometry may be examined and refuted most easily, certainly, 
and justly by arithmetical computation. Also all true conclu- 
sions and lawful demonstrations in geometry may be con- 
firmed and illustrated by the help of an arithmetical calculus, 
which consideration being presupposed and established, the 
method of treating geometry will be found nowhere to differ 
from the calculations of arithmetic. 

[Barrow's Mathematical Lectures, Cambridge, 1734. 

A number is emphatically a measure. Numbers arc as 
much the means and the subject of measure as quantities. 
This presupposes that the unit is known. Thus, one inch, one 
mile, one pound, one ton, are units of measurement, and where 
their quantity is known as units, they give the power of 
counting, of computing, of measuring. If we know that the 
English ale gallon, usually taken as our water gallon, contains 



38 DEFINITIONS. 

282 cubic inches, and that one cubic foot contains very nearly 
6 j$y of such gallons — if we know the relations of these quanti- 
ties as units, they become the means of estimating and deter- 
mining any multiples of each other in any reservoir we may 
wish to measure. 

Measurements after all are only comparisons ; they denote 
nothing but that a magnitude is capable of being compared 
with other- magnitudes, so that its quantity, however other- 
wise unknown and indeterminate, is capable of being known 
and determined from the relation which it obtains to them. 

We require for measurement, standards mutually recog- 
nized. Thus, an Englishman who was ignorant of the French 
metre, would not derive any idea from it. He must translate 
or compare it with the English yard, or English inch, before 
he could have any accurate notion of the distance meant. 

We have defined Mathematics to be the science of measure- 
ment. Measurement, in a comprehensive sense, including 
multitude and magnitude, force, motion, time, weight, &c. 
Measurements are either natural or artificial. A measure- 
ment, philosophically speaking, is only a comparison ; it is a 
comparison with some known standard, either natural or 
conventional. Thus, a day, considered as the period of the 
earth's revolution on its axis, is an invariable natural measure- 
ment of time, the same in all ages and in all parts of the earth. 
We divide the day artificially into 24 hours of 60 minutes — 
1440 minutes, and 86,400 seconds. These are artificial, con- 
ventional measurements. We might divide the day into ten 
divisions, or any other number, but these would in any case 
be equally arbitrary sub-divisions of an invariable natural 
standard, the day. A number of days taken together, form 
other conventional divisions, as a week, a fortnight, a month. 
The year, the period of the earth's revolution round the sun, 
causing the Seasons, is another natural standard of time. 



DEFINITIONS. 3D 

Thus, we make days the measure of the period of revolu- 
tion round the sun of the nearer Planets, and years the mea- 
sure of the more distant ; the most distant, the latest discov- 
ered planet, Neptune, occupying hundreds of years. 

An inch, a foot, a yard, a bushel, a pound, are artificial 
measurements of length. We require these conventional mea- 
surements. We could not measure without them. 

What is an inch'? The United States being originally 
colonies from England, our measurements are those of the 
English standard. Thus, our inch is the English inch, one 
twelfth part of a foot, one thirty-sixth part of a yard. 

The English yard was originallv taken from the length of 
the arm of Henry I, King of England, and was afterwards 
confirmed by Acts of Parliament, from the reign of Henry 
III to George IV, in 1824. 

The original yardstick is carefully preserved, we believe, in 
Guildhall, London, to compare others with, so as to have them 
all alike, and it is necessary to have officers whose duty it is 
to see that yardsticks and all other measures and weights are 
properly regulated. 

There is an officer at Washington, whose duty it is to have 
correct weights and measures made for standards, and these 
arc distributed to the different States and cities. There is 
also in most cities and large towns a Regulator of Weights 

o o o 

and Measures, whose duty it is occasionally to examine the 
weights and measures in actual use. 

The French metre is a sub-division of an invariable natural 
standard. It is one ten millionth part of the distance from 
the Equator to the Pole. It is equal to 39-^l English 
inches. 

Natural measurements are those not subject to variation 
from place, time, or chance, being the same invariably at all 
times, and in all parts of the earth. It is the purpose of the 



40 DEFINITIONS. 

science of mathematics to contemplate all measurements 
whether natural or artificial, and all contrivances which have 
been devised for comparing their relations to each other. Not 
only the relations of length, breadth, and thickness [distances, 
surfaces, masses] but also of forces, motions, velocities, inten- 
sities, times, weights, and the various relations and comparisons 
of all and each of these, interchangeably with each other. 

" In natural measurements we call to our assistance the pre- 
sumed permanence of the great laws of nature with all 
experience in its favor, and the strong impression we have of 
the general composure and steadiness of every thing relating 
to the gigantic mass we inhabit — " the great globe itself." In 
its uniform rotation on its axis, accordingly, we find a stand- 
ard of time, which nothing has ever given us reason to regard 
as subject to change, and which, compared with other periods 
which the revolutions of the planets about the sun afford, has 
demonstrably undergone none, since the earliest history. 

"In the dimensions of the earth we find a natural unit of the 
measure of space, which possesses in perfection every quality 
that can be desired ; and in its attraction combined with its 
rotation, the researches of dynamical science have enabled us 
through the medium of the pendulum, to obtain another inva- 
riable standard, more refined and less obvious, it is true, in its 
origin, but possessing a great advantage in its capability of 
ready verification, and therefore easily made to serve as a 
check on the other." 

[Sir John HerscheVs Natural Philosophy. 

" Algebra, n. s. [an Arabic word of uncertain etymology ; 
derived, by some, from geber, the philosopher; by some from 
gefr, parchment ; by others from algehista, a bone-setter ; by 
Menage from algiatarat, the restitution of things broken.] 

" This is a peculiar kind of Arithmetic, which takes the 
quantity sought, whether it be a number or a line, or any 






DEFINITIONS. 41 

other quantity, as if it were granted, and, by means of one or 
more quantities given, proceeds by consequence, till the quan- 
tity at first only supposed to be known, or at least some power 
thereof, is found to be equal to some quantity or quantities 
which are known, and consequently itself is known. 

" The origin of this art is very obscure. It was in use, 
however, among the Arabs, long before it came into this part 
of the world; and they are supposed to have borrowed it 
from the Persians, and the Persians from the inhabitants of 
India. * * * The first Greek author of Algebra 
was Diophantus, who, about the year 800, wrote thirteen 
books. — [Dr. Johnson's Folio Dictionary. 

" Algebra is a branch of the mathematics, which has for its 
object whatever can be expressed by numbers, either exactly, 
or by approximation. 

"In this respect, and also in its employing arbitrary signs to 
denote the things of which it treats, it agrees with arithmetic. 
The analogy between the two sciences induced Sir Isaac 
Newton to denominate it Universal Arithmetic ; but by the 
application of algebra to geometry, the science has acquired a 
new character and new powers, which render this appellation 
too limited, and not sufficiently descriptive of its nature. In 
its present state, it is nearly alike related to arithmetic and 
geometry. In its application to both sciences, the reasoning 
is carried on by general symbols ; its true character consists 
in this, that the results of its operations do not exhibit the 
individual values of the quantities which are the subject of 
investigation, such as we obtain in arithmetical or geometri- 
cal constructions. They only indicate the operations, whether 
arithmetical or geometrical, which ought to be performed on 
the given quantities, to obtain the value of the quantities 
sought. 

[Encyclopedia Brittannica, 1th edition. 



42 



DEFINITIONS. 



"Algebra, Buchstaben-rechnung : letter calculation, alge- 
braic reckoning ; from Buchstaben, a type ; letters applied to 
the designation of magnitudes." — [German Die. 

" Algebra, n. s. from the Arabic ; the reduction of parts to a 
whole, or fractions to whole numbers ; from the Arabic verb, 
which signifies to consolidate." — [Webster's, Die. 1828. 

" Analysis in mathematics, a method of solving or resolving 
mathematical problems. 

" There are two general methods of finding truth in mathe- 
matics — Synthesis and Analysis. Synthesis is the composi- 
tion, or the putting of several things together, (the building up 
as it were, of truth ;) the method of Analysis consists more in 
judgment and readiness of apprehension, than in any particu- 
lar rules. It furnishes the most perfect instances and exam- 
ples of the art of reasoning. By it, geometrical demonstra- 
tions may be wonderfully abridged. 

[Chambers' Encyclopedia, London, 1743. 

" Analysis, n. [avalvaig, of ava and lvci5,~\ a loosing or resolv- 
ing, from Uco, to loosen. 

1. " The separation of a compound body into its constituent 
parts; a resolving, as an analysis of water, air or oil, to 
discover its elements. 

2. " A consideration of any thing in its separate parts ; an 
examination of the different parts of a subject, each sepa- 
rately, as the words which compose a sentence, the notes of a 
tune, or the simple propositions which enter into an argument. 
It is opposed to synthesis. 

" In mathematics, analysis is the resolving of problems by 
algebraic equations. The analysis of finite quantities is other- 
wise called Algebra or specious Arithmetic. The analysis of 
infinites, is the method of Fluxions, or the Differential cal- 
culus. 



DEFINITIONS. 



43 



" In logic, analysis is the tracing of things to their source, 
and the resolving of knowledge into its original principles. 

3. " A syllabus or table of the principle heads of a continued 
discourse, disposed in their natural order. 

4. " A brief methodical illustration of the principles of a 
science. In this sense it is nearly synonymous with synopsis." 

[Noah Webster, 1828. 

"Analysis in mathematics is properly the method of resolving 
problems by means of algebraic equations, whence we often 
find that these two words, analysis and algebra, are used as 
synonymous. 

"Analytic, or analytical, something that belong to or par- 
takes of the nature of analysis. The analytic method stands 
opposed to the synthetic." 

[Stewart's Philosophy, vol. 2 chap. 4, sec. 3. 

Algebra is a contrivance by which letters or symbols are 
used to represent quantities, and their relations to other quan- 
tities. Algebra is either Arithmetic or Geometry. It is 
applied to whatever can be otherwise stated as questions of 
Arithmetic or Geometry. It is a method of short hand, 
very convenient, and often nearly indispensable, as a means 
of recording the processes of Arithmetic and Geometry. 

Algebra enables us to express a general truth, independent 
of any particular application of it. Sir Isaac Newton desig- 
nated it " Universal Arithmetic." But this does not seem a 
very satisfactory definition. We have many exercises in 
geometry and arithmetic thus expressed by algebraic symbols, 
and the utility and convenience of these symbols are undoubted, 
if used in their proper and true place. But the authors of 
these short hand methods have been so numerous, and their 
symbolic language so various, that the ingenuous beginner, 
and even the ingenious studenl who Ims made considerable 
progress, is often hopelessly fatigued and mystified by them. 



44 DEFINITIONS. 

Even so early as 1728, the publisher of the second corrected 
edition of Sir Isaac Newton's " Algebra, or Universal Arith- 
metic" (first published in 1707,) says: "It is true we have 
already a great many books of Algebra, and one might even 
furnish a moderate sized library purely with authors on that 
subject." But Sir Isaac's own methods are not models of 
perspicuity. 

Prof. Barrow, in his mathematical lectures at Cambridge, 
1734, page 28, (it may be remarked that he could not be 
ignorant of Sir Isaac Newton's work, published six years 
before,) says : " Perhaps some may wonder, that while I am 
endeavoring to make a perfect enumeration of all the parts 
(at least the principal parts) of Mathematics, I am wholly 
silent about that which is called Algebra, or the Analytic art. 
I answer that this was not done unadvisedly ; because, indeed, 
Analysis, understood as intimating something distinct from the 
rules and propositions of .geometry and Arithmetic, seems to 
belong no more to Mathematics than to Physics, Ethics, or 
any other science. No more is Synthesis, which is the manner 
of demonstrating theorems in contradistinction to Analysis." 

Dugald Stewart, (vol. 2, chap. iii. sec. ii,) remarks that it 
may be questioned whether, among the ancient Greek geome- 
ters, this power, the power of reasoning, was not in a higher 
state of cultivation, in consequence of their ignorance of the 
algebraic symbols, than it exists in at this day, among the 
profoundest mathematicians of Europe. 

The same distinguished philosopher devotes twenty pages, 
(vol. 2, chap. iv. sec. iii.) in which he most distinctly and 
formally dissents from Newton's statement of the priority and 
importance of what are designated the analytic methods, and 
closes with the following paragraph : 

" It were surely better that mathematicians should cease to 
give the sanction of their authority to this misapplication of 



DEFINITIONS. 45 

the words analysis and synthesis, as it has an obvious ten- 
dency, beside the injustice which it involves to the inestimable 
remains of Greek geometry, to suggest a totally erroneous 
theory, with respect to the real grounds of the unrivalled and 
transcendent powers possessed by the modern calculus, when 
applied to the more complicated researches of physics." 

The truth is, and ought to be admitted, that a synthetic 
process in Arithmetic or Geometry, equally with an analyti- 
cal one may be at pleasure expressed by algebraic symbols. 

Algebra is only short hand ; as such, it is equally applicable 
to synthetical or analytical processes. To speak of it, there- 
fore, as an analytical method, in contradistinction to a syn- 
thetical, is both false and absurd. 



CHAPTER III. 

ON THE NATURE OF DEMONSTRATIVE EVIDENCE. 

In the first chapter sufficient authorities have been quoted 
to establish the utility and the importance of mathematical 
studies as an indispensable branch of popular education. 
Mathematics is a science in which, however, authorities 
are of less importance than in any other. In mathematics, 
self-evident conviction, founded upon observation and experi- 
ment, is alone expected and relied upon to compel and com- 
mand the assent of each individual. The opinions, or rather 
the researches of others have indeed great value to us, but 
only as they may be guides to the action of our own minds. 

The present state of mathematical knowledge is a very 
slowly accumulated result of the labors of many contributors 
during a long series of ages. Among these are several men 
whose genius and patient industry has not been surpassed by 
the votaries of any other pursuit. 

Yet, although the science is admitted to be the most re- 
markable monument of the human intellect, its methods of 
teaching, classification, and nomenclature appear to require 
revision, and it is the purpose of this chapter to offer some 
reason or apology for an attempt to simplify and to render 
attractive this great subject. 

The proportion of students who succeed in acquiring a 
comprehensive knowledge of mathematics by the existing 
methods is surprisingly small, in any country. Our purpose 
is to consider whether the very limited number of persons who 
now derive benefit from its teaching is the result of difficulties 
inherent in the subject itself, or of the inadequate and ill- 
adapted methods of instruction which are generally adopted. 
46 



ON THE NATURE OF DEMONSTRATIVE EVIDENCE. 47 

I shall here offer the opinions of others rather than extend 
my own observations. 

I have before me " Observations on the Nature of Demon- 
strative Evidence, with an Explanation of certain difficulties 
occurring in the Elements of Geometry," by Thomas Beddoes, 
written at Oxford, and published St. Paul's Churchyard, 
London, 1793. 

Its author, Dr. Thomas Beddoes, was Professor of Chemis- 
try at Oxford, and subsequently was known as a distinguished 
chemist and physician in London, and contributor to medicine 
and chemistry, at a remarkable epoch of chemical discovery, 
when Black and Priestly were in the zenith of their labors. 
The work is dedicated to, and is sanctioned by the appro- 
bation of, Davies Gilbert, at that time Professor of Mathe- 
matics at Oxford, and subsequently President of the Royal 
Society. Prof. Gilbert was a gentleman of uncommon pro- 
ficiency, and rare discernment in mathematical studies. It 
may be also remarked that Sir Humphrey Davy, perhaps the 
most distinguished chemist, and lecturer on Chemistry which 
England has produced, was a pupil of Dr. Beddoes, having 
been commended to him when a youth by Prof. Gilbert. 

Dr. Beddoes had extraordinary merits as an original thinker. 
He very strongly approves of, and enforces the views of 
Locke. In his dedication he says : 

" The more I consider the subject, the more I am inclined, 
in spite of Mr. Harris, to believe not only in the possibility, 
but the utility of rendering^the elements of geometry palpable. 
If they be taught at an early age, a plan in which I think I 
see many advantages — models would make the study infinitely 
more engaging. From the mere slate and pencil most begin- 
ners experience a repulsive sensation. But if a child had 
something to handle, and to place in various postures, he 
might learn many properties of geometrical figures, without 



48 



ON THE NATURE OF 



any constraint upon his inclinations. He would have no 
difficulty in transferring the properties of palpable, to merely 
visible figures, nor in. generalizing the inferences. You will 
not object that one cannot proceed far by this road : you will 
perceive that much more would be gained in reality than 
appearance. We should have laid a good foundation for the 
invaluable habit of accurate observation in general ; and 
towards future progress in mathematics, we should have 
warded off the first disagreeable impression of the aspect of 
the science, which is so very apt to strike a damp to the 
heart of the beginner. 

" I need not explain to you the advantage of trying to 
engage Fancy on our side by all the allurements we can offer 
to her. It is she that smooths every path, and strews it with 
flowers. We all, men and boys, follow with alacrity wher- 
ever she leads ; neither the mind nor the body grudge any 
labor ; and it is the enthusiasm she inspires that has worked 
so many miracles in art and science. By some strange 
fatality, however, she is neglected, if not affronted, in almost 
all the stages of education ; and the first step in almost every 
species of instruction is to present knowledge to the student's 
imagination in conjunction with some melancholy and hateful 
accompaniment; which sort of management I conceive to 
have much the same kindly influence upon this faculty, as an 
unseasonable frost upon the tender petals of an expanding 
blossom. 

" The mode of initiation in geometry which I propose, could 
not, unless I very much deceive myself, fail to render the im- 
pressions of sense more agreeable by rendering them more 
distinct. The rigorously scientific method, as it is supposed 
to be, seems, on the contrary, to aim only at rendering them 
as obscure as possible ; an intention, I confess, perfectly in 
unison with the other parts of the established process of school 
and college stupefaction. 



DEMONSTRATIVE EVIDENCE. 49 

" Whether you will allow that this important point is likely 
to be in this manner attained, I am not sure. But you will 
agree with me in thinking that it is high time to discard 
Euclid's Elements. The science cannot be exhibited in a 
more disgusting form, as we may be convinced by daily 
instances. Nor are these Elements any way necessary to lay 
a good foundation in mathematics, for there are few I will 
venture to guess, of the eminent mathematicians of Europe 
that have been initiated by the study of Euclid. 

" That, by laying Euclid aside, we should be deprived of 
what Bacon calls the intervenient advantages of mathematics, 
is to me a vain apprehension. Those who have dragged 
their understanding laboriously along the tiresome circuit of 
ancient demonstration, may be unwilling to grant that they 
have taken all these pains to no purpose. Yet they can 
hardly flatter themselves in secret that they have acquired 
habits of attention or abstraction superior to those who having 
pressed forward by the nearest road, enjoy both the direct and 
indirect profits of their labor. 

" For want of time, or opportunity, or resolution, I am not 
able to take so comprehensive a view of the subject as I could 
wish, I cannot indeed suppose it possible that either branch of 
mathematics should ever cease to be ' Quails ab incepto pro- 
cesserit,' but I am apprehensive that I have not presented my 
observations in so advantageous a form as I might have done 
if I had possessed more knowledge. I hope, however, to be 
intelligible to those who may choose to examine them. In the 
main principle I cannot suppose myself mistaken; where I 
have committed errors in the application I shall receive cor- 
rection with the cheerfulness becoming a person equally ready 
to hear and to tell the truth. 

I am, with sincere regard, yours, 

THOMAS BEDDOES." 
Oxford, September 6, 1792. g 



50 ON THE NATURE OF 

We shall select a few detached portions, the heads of a few 
paragraphs, from the text, which may give some idea of its 
aim and purport. 

Page 1. "In proportion as the writings of Mr. Locke rose 
in the public esteem, * # 

Page 2. " Mr. Locke's success has, I imagine, already 
contributed, and will hereafter in a greater degree still con- 
tribute ####### 
one may, without impropriety, call a book popular that has 
gone through twenty editions. 

Page 5. " In whatever study you are engaged, to leave 
difficulties behind is distressing ; and when these difficulties 
occur at your very entrance upon a science professing to be 
so clear and certain as geometry, your feelings become still 
more uncomfortable, and you are dissatisfied with your own 
powers of comprehension. 

Page 10. " The pretensions of the abstract sciences have, it 
must be confessed, something wonderfully alluring. 

Page 13. "In mathematical reasoning, the mind grasps the 
conclusions with full assurance of their reality ; we are satis- 
fied that our advances in this science are actual acquisitions,, 
and we find them as we go on continually capable of applica- 
tion. 

" It may therefore be interesting to inquire into those cir- 
cumstances which constitute the irresistible force of mathe- 
matical evidence. We shall, at the same time, if we are 
successful in this inquiry, discover upon what depends the differ- 
ence in the cogency of proof between demonstrative evidence, 
and such evidence as less powerfully commands our assent. 
Without this, I do not see how we can ever take a clear 
survey of evidence in general, or enjoy the satisfaction of 
accounting to ourselves fully for our own conviction or belief. 

" It seems to me, in the present state of our knowledge, so 






DEMONSTRATIVE EVIDENCE. 51 

easy to point out the nature of this and the other sorts of 
evidence, that I wonder how it can be mistaken. Yet fre- 
quently as the topic is expatiated upon, I know no book in 
which the true principles have been fully explained and 
applied ; and in general, I have reason to believe that very 
erroneous ideas prevail upon a subject of unquestionable im- 
portance to the theory of the human understanding;. # * 

Page 15. "On examining a train of mathematical reason- 
ing, we shall find that at every step we proceed upon the 
evidence of the senses ; or, to express myself in different 
terms, I hope to be able to show that the mathematical 
sciences are sciences of experiment and observation, founded 
solely upon the induction of particular facts, as much so as 
mechanics, astronomy, optics or chemistry. In the kind of 
evidence there is no difference, for it originates from percep- 
tion in all these cases alike, but mathematical experiments are 
more simple, and more perfectly within the grasp of our 
senses, and our perceptions of mathematical objects are 
clearer. ####### 

Page 16. "No sooner do we look into an elementary trea- 
tise for the proofs of this opinion, than we meet with them at 
every step, in every demonstration ; and I shall, I hope, be 
allowed to have established it firmly, if I shew that Euclid 
sets out from experiments, and proceeds onwards by their 
aid, appealing constantly to what we have already learned 
from the exercise of our senses, or may immediately learn. 
The same thing must needs be equally true of every other 
elementary author. After having exemplified the nature of 
demonstrative reasoning, I shall leave the reader to extend 
this mode of considering it to other cases, in a full persuasion 
that he will find the same process repeated in every demon- 
stration upon which he may choose to make the trial. * * 
He here gives examples. * * * * * 



52 ON THE NATURE OF 

Page 24. " I have been purposely prolix in this demonstra- 
tion, to show how it begins in experiment, goes on by experi- 
ment, and ends in an experimental conclusion. There may be 
another use in insisting so particularly upon the nature of the 
reasoning process here : among those who teach mathematics 
without understanding their practical application, and also 
Without entertaining a just idea of the nature of demonstra- 
tion, there prevails a sort of pedantry, productive of infinite 
disgust to the learner. If by detached figures I could show 
the truth of any proposition in an instant, I am forbidden, 
because this is an unmathematical mode of proceeding ; that 
is, mathematical reasoning is supposed to be something inde- 
pendent of experience, and the science to be more refined 
than the experimental sciences. Hence, if a Greek writer 
happens to have written a demonstration a mile long, which 
demonstration can be nothing but a concatenation of the 
results of observation and experiment, I must take this tedious 
round, rather than be allowed to arrive at the point desired 
by only traversing half a dozen yards, provided this shorter 
road leads through the unhallowed region of the senses. * * 

Page 27. " In this manner does every demonstration pro- 
ceed upon the results of experiments, as the reader will find, 
in as many instances as he shall take the pains to examine. 
And since the appeal in demonstrative reasoning is always 
made to what is now exhibited to the senses, or to what we 
have before learned by the exercise of the senses, too much 
pains cannot be taken at the commencement of the study of 
geometry, to satisfy the mind of the learner, by appealing to 
his senses. The more distinct and deep the impressions of 
sense are at the beginning, the greater will the power of 
abstraction afterwards be, when the progress of his studies 
shall have carried him into the higher mathematics. Abstrac- 
tion is not in fact a distinct power, as the metaphysicians, 



DEMONSTRATIVE EVIDENCE. 53 

who seem to imagine that they increase the importance of 
their science, as they multiply distinctions, teach # * 

Page 29. " By appealing in this manner to his senses, and 
making him feel the firmness of the ground on which he 
treads, one might probably instruct a boy, at an early age, in 
the elements of geometry, so as rarely to give him disgust, 
and frequently great satisfaction. He would, by impercepti- 
ble degrees, acquire the power of abstraction, or learn to 
reconsider each separate perception, as well as to combine 
them anew." 

Dr. Beddoes next devotes a number of pages to the defini- 
tions and axioms used in Euclid, and other elementary books 
of geometry, and quotes Locke, Book IV. Chapter 7, Section 
8, where Locke shows conclusively that these axioms are not 
the fundamental and primary truths first known to the mind, 
and that the other parts of our knowledge do not depend upon 
them, and also that these axioms are by no means the only 
self-evident truths, but that self-evident truths are in fact very 
numerous. He also shows that general ideas are formed 
slowly from the consideration of particular ideas with which 
we must be first familiar. 

Page 38. " All accurate ideas founded upon measurement 
or careful comparison, must necessarily be subsequent to 
approximations suggested by a casual or distant survey of 
objects ; and a good deal of perplexity, when persons first 
engage in the study of mathematics, arises from the result of 
vague observations, mixing itself with the result of the exact 
experiments employed in the reasoning of mathematics." 

Page 58. " In Euclid's Elements, the truth seems to me to 
be so frequently obscured by demonstration, and so much 
disgust is often excited by his tedious method of proceeding, 
that were it hot a violation of that loyalty which we owe to 
our masters, the Greeks, I wish the shortest possible method 



54 



ON THE NATURE OF DEMONSTRATIVE EVIDENCE. 



might be followed in teaching the rudiments of mathematics, 
by the help of simple satisfactory experiments. And if there 
be any one who should have learned geometry in this way, 
let him be assured that he holds his proficiency by a firm 
tenure. In this science there is no transcendental road; 
but I imagine a royal road might be struck out, though 
Euclid was of a different opinion. # # # # 

Page 88. " Clear as it is, no use has I believe, yet been 
made of Mr. Locke's account of demonstrative evidence, 
either to solve difficulties or to improve the method of teach- 
ing in geometry : the shame, however, will be divided between 
so many culprits, and some of them so illustrious, that the 
share of each will be exceedingly small. For the subject has 
not fallen into the hands of ordinary compilers merely. Among 
the commentators upon Euclid, one might enumerate men of 
comprehensive views and various information ; and authors, 
who aspire far beyond the commentator's highest praise, have 
formally discussed the nature of mathematical evidence. # # 

Page 114. "Mathematics, in fine, teach either to measure 
or to count. The simplest, and the shortest way we can 
acquire either of these arts, the better, I believe, in all respects. 
We cannot possibly set about to learn either of them other- 
wise than by the use of the senses. And it is by no means 
impossible, that there may be a method of applying the 
senses, in geometry at least, to far greater advantage than 
any practised at present ; a method at once agreeable, expe- 
ditious, and calculated to invigorate every mental faculty. 
When such a plan of education shall be generally adopted, 
(and its outline is not extremely difficult to trace) that more 
knowledge than the learned and scientific now usually possess 
at forty, may be acquired by twenty, such a method of 
teaching geometry will probably form part of it.'* * * 



CHAPTER IV. 



EXPLANATION OF THE PLATES. 



The science of measurement is capable of three great pri- 
mary divisions. The first comprises measurements of one 
dimension only, — length. We can contemplate a length, a line, 
a distance of any given length. We can add to it ; we can 
subtract from it ; we can multiply its length till it may exceed 
the distance from the earth to the sun, or to the remotest of 
the planets ; or any assignable length. Length is thus capable 
of almost infinite extension. We can also divide any given 
distance, any length, however great, and may continue to 
divide it, till it becomes exceedingly short, microscopically — 
mathematically small. But it is still length only. It is a line 
— a distance — a measurement of one dimension. It does not 
contemplate either breadth or thickness. 

The second great division comprises measurements of two 
dimensions, length and breadth. These are planes — surfaces 
areas — -superficial measurements only. Of this class of mea- 
surements the invariable law is the multiplication of the length 
by the breadth, the addition of the breadth to the length, (for 
multiplication is but a variety of addition,) which will in every 
case be true. 

The third class of measurements are those of three dimen- 
sions, length, breadth, and thickness. Of these, the invariable 
law is multiplication, first of the length by the breadth, and 
secondly multiplying the product thus obtained by the third 
dimension, thickness, giving us cubical or solid measurements. 

There is no fourth variety of measurement in nature, there- 
fore every question of measurement, from the simplest to the 
most complex, must belong to one or to the other, or to 
combinations of these three. There can be no other. 

55 



56 EXPLANATION OF THE PLATES. 



PLATE No. 1. 



The purpose of this diagram is to exhibit in the most simple 
manner the three varieties of measurements (or dimensions) 
which exist in nature. 

On the left. Figure 1 ; there are three lines of one inch in 
length, of differing breadth or thickness, but all of them 
equally lines, for a line may be indefinitely thin. There are 
also above these, similar lines of two, three, and four inches, 
and of twelve inches or one foot in length. 

A line represents a distance, one dimension — length, for 
instance. It is not necessary that a line should be actually 
marked. A line run by a surveyor of land is usually marked 
by slicing the bark of trees adjacent to the line, but it is the 
corners which are chiefly relied on. A line of latitude, or the 
equinoctial line, where it crosses the earth or ocean, is mea- 
sured from the heavens and the sun's apparent path ; it is not 
traced upon the earth. A distance across a river, or an arm 
of the sea, is likewise accurately measured without the actual 
line, by triangulation. 

Though we speak of a line, no line is actually drawn. It is 
an imaginary line,. an air line, a line which may be measured, 
but it is not marked. In measuring a very short distance, as 
the width of a street or of a city lot, we may perhaps be said 
to draw a line when we stretch a tape across it. A line, 
as it represents length only, is often said to have no breadth ; 
but a line as drawn upon paper if it had no breadth would not 
be visible. 

A line, then, is a distance, one dimension ; it may be from 
the earth to the sun, or to the most distant of the planets. We 
may conceive a line of any length, however great, and we 
may continue to add to it, or to multiply this length until it 
extends almost to infinity, (in other words, longer than any 



PLATE NO. 1. 57 

definite length.) We may then divide it, and may continue to 
divide it until it becomes indefinitely short, microscopically 
short, but it is still a line, however short the distance. 

In Figure 2, we have an example of the second class of 
measurements, those of two dimensions, length and breadth. 
It is one inch in length, and one inch in breadth. We say 
once one is one. The square of one is one. One length 
multiplied by one breadth is one, (one unit of the second 
class of dimensions,) one area, one surface, one superficial 
measurement. We have also in Figure 3 two inches length 
by two in breadth, the area being the simple multiplication of 
the length by the breadth. In Figure 4 we have three inches 
by three inches, and in Figure 5 twelve inches by twelve 
inches, equal to 144 square inches, or one square foot. In 
every possible case of measurements of two dimensions, the 
rule is invariably the multiplication of the length by the breadth. 

Figure 6 represents an example of the third class of mea- 
surements, those having three dimensions, length, breadth, and 
thickness. Here we multiply, first, the length by the breadth ; 
then, the product of the length and breadth by the third 
dimension, the thickness, giving solid measurements — the 
cube. Once one is one, one length multiplied by one breadth 
is one, (one superficial or square inch.) Once one is again 
one, one length and one breadth is now multiplied by one 
thickness, (one cubic inch.) 

In Figure 7 we have twice two are four, twice four are 
eight. Eight cubic inches is the cube of two. 

In Figure 8 we have three times three are nine, three times 
nine are twenty-seven. Twenty-seven cubic inches is the 
cube of three. 

To form the cubic foot from the square foot, as is repre- 
sented in Figure 5, we multiply 144 by 12, giving 1728, the 
number of cubic inches in the cubic foot. 



58 EXPLANATION OF THE PLATES. 

Great advantage to the beginner will be found to result 
from a tangible demonstration by handling a number, say as 
many as convenient, of cubic inch blocks of mahogany or any 
hard wood, and thus forming the cubes for himself. 

The use of this diagram, as already stated, is simply to 
exemplify and point out length, breadth, and thickness, the 
primary elements of mathematics. To one or the other of 
these three relations, every possible mathematical question 
necessarily belongs. 

We repeat, that every possible mathematical question, from 
the simplest which can be offered to the infant mind, to the 
most complex which can ever be devised by the ingenuity of 
man for the purpose of puzzling and of mystifying his fellows 
in exercises of calculation, must be founded upon one or the 
other, or upon combinations of these three elements. In 
Nature there is no fourth variety of measurement, no fourth 
power ; in the language of arithmetic there are fourth, fifth, 
and higher powers, but these may be said to have no real 
existence ; they have no application to forms or motions in 
Nature, and consequently are only mere exercises of calcula- 
tion in Arithmetic or Algebra. 

Perhaps the most important primary lesson which can be 
taught to the student of mathematics is the natural and direct 
connection between these three classes of measurements, with 
all the varieties and all the relations of motion. Every motion 
is a result of force. The nature or origin of force we cannot 
always understand; but we can observe and measure the 
motion, of which it is the cause. 

One force produces motion in a straight line, and only in a 
straight line. Every motion which is in a straight line is a 
result of one single force, or it may be, one aggregate force. 
It represents, and is capable of being measured by the first 
class of measurements. 



PLATE NO. 2. 59 

A motion which is not in a straight line must always be a 
result of more than one force. Thus a stone thrown into the 
air returns to the earth. It has a curved linear motion. The 
one, (say the projectile,) force would have impelled it in a 
straight line, but gravity, a second force, acting upon it per- 
pendicularly, brings it to the ground. In like manner, the 
movement of a stone attached to a string held in the hand 
describes a curved linear motion. So also do the revolutions 
of the planets round the sun, and of their satellites round the 
planets. These motions are in a plane. It has length and 
breadth. It is not one mere line. It possesses the properties 
of, and is capable of being measured by the second class of 
measurements. 

.The third class of motions are those of radiant phenomena, 
acting from a centre on all sides, as the diffusion of light, heat, 
gravity, electricity, sound, &c. All radiant phenomena illus- 
trate each other. They diffuse themselves on all sides ; they 
have length, breadth, and thickness. They belong to the 
third variety of measurements. As there are no more than 
three measurements of space and quantity in nature, so there 
are but three varieties and measurements of motion in the 
universe. 



PLATE No. 2. 

Every square and every parallelogram possesses the pro- 
perty of having its sides parallel to each other. 

This Plate is designed to exemplify, First, the fact that 
every square and every parallelogram is capable of being 
divided diagonally into, and may therefore be considered as 
consisting of two similar and equal triangles. 

Second. That this is true and invariable, not only of the 
right-angled parallelograms, but of those which are oblique. 



60 EXPLANATION OF THE PLATES. 

Third. That the areas of triangles, like the areas of squares 
and of parallellograms, which are in fact only double triangles, 
are a simple result of the multiplication of their length by 
their breadth. 

In Figure 1, we have an example of right-angled parallelo- 
grams of equal sides, that is, squares. Figures 2 and 3 are 
not right-angled ; they are oblique and more oblique ; and are 
called rhombs, but each of them may be divided into two 
equal triangles. 

Figures 4, 5, and 6 are similar examples of parallelograms. 
Several diagonal lines are drawn, dividing these into equal 
and similar triangles. 

Figures 7, 8, 9, 10, and 11, though the length and breadth 
are unequal, and the size and form and angles different, are 
further examples of this truth ; which will be found on experi- 
ment to be invariable, and to apply without exception to 
every parallelogram, and the reverse, that by adding to any 
triangle a similar triangle of equal size, we should form a 
parallelogram, or double triangle, is obvious. 

Figure 12 exhibits the truth that any triangle, if we double 
its length and breadth, quadruples its area, precisely the same 
as any parallelogram or double triangle. We may start from 
the left or the right, or from the apex, it does not alter the 
fact. Figure 12 is at its top an angle of 120 degrees, or one 
third of 360 degrees, into which the circle is divided. 

Figures 13, 14, 15, 16, are respectively at the top, angles 
of 90 degrees, 72, 60, and 45 degrees, (I, 1, I, and 1, of 360 ;) 
they are successively more acute, but the increased acuteness 
does not alter the rule. It applies to every triangle, without 
exception. 

Figure 17 is two triangles (of 30 degrees,) touching at the 
apex. On the right, the length and breadth are doubled, pro- 
ducing as in the previous examples, four similar triangles. 



PLATE NO. 3. 61 

On the left, the length and breadth is subdivided, and if, 
starting from the apex, or from either angle, we triple and 
quadruple the length and also the breadth, we would form 
successively, nine or sixteen similar and equal triangles, and 
so on. It is obvious, that if to the first triangle, or to any 
triangle whatever, we had before added another similar 
triangle (in reverse,) and thus made it a parallelogram, and 
had then quadrupled its length and breadth, we should have 
formed sixteen similar parallelograms or double triangles. 
The rule applies invariably and universally to all forms, as we 
must in due time make evident. It will be found to apply 
equally to all figures, whether bounded by straight lines, or 
by curved or crooked lines. It is the universal law of all 
superficial measurement to multiply the length by the breadth. 
The phrase is a common one, to say that an area, a quan- 
tity, or a force, increases or diminishes as the " Squares of the 
Distances." This means simply and nothing more than the 
length multiplied by the breadth. It would often be equally 
proper, and more true, to say that it increases as the circles 
of the distances, or as the triangles of the distances. Thus, 
the left hand portion of Figure 17, starting from the apex, 
well illustrates the law of falling bodies — gravity, &c. 



PLATE No. 3. 

This diagram makes obvious to the eye the truth that all 
length and breadth justly considered are at right-angles with 
each other, and that if planes or solids are extended obliquely, 
drawn out, between the same parallels, the area is, the mass 
is still invariable, however great the extension. For a famil- 
iar example, if you draw out a piece of india rubber, or of 
clay, or of dough in length, it diminishes in breadth in the 
same proportion ; in reality it is exactly the same quantity. 



62 EXPLANATION OF THE PLATES. 

This is perhaps, the simplest primary idea of fluent quantities, 
quantities flowing, quantities drawn out, water flowing, the 
relation of forces (and their result motions) to times, spaces, 
&c, in short, of many mathematical measurements, called by 
a variety of names ; as Fluxions, Analysis of Infinities, &c. 

Figure 1 is a parallelogram of 2 inches by 1 inch, say 2 
inches length by 1 inch in breadth. As is exemplified in 
Figure 2, it is capable of division into two equal and similar 
triangles. In Figure 2 we have the diagonal line drawn, and 
the two triangles formed ; and in addition a third similar 
triangle is drawn on the right of the parallelogram. The two 
triangles on each side of the central triangle are equal to it, 
and equal to each other. That on the left, together with the 
central one, forms a right-angled parallelogram — a rhomb — 
obviously of equal area. 

In Figure 3, we have again the same right-angled parallelo- 
gram as in No. 1, subdivided into four triangles. To this, on 
the right are added three triangles, which, with the one triangle 
which belongs to both, forms a parallelogram of the same 
area, and of twice the obliquity of that of Figure 2. 

In Figure 4 we have again the same right-angled parallelo- 
gram as in No. 1, but subdivided into six triangles. To this 
on the right are added five triangles, which, with the one 
triangle which is common to both, forms a parallelogram of 
the same area, but of triple the obliquity of No. 2. 

In Figure 5 we have the same right-angled parallelogram, 
but subdivided into eight triangles. To this on the right are 
added seven triangles, forming with the one which is common 
to both, a parallelogram similar in area, but four times as 
oblique as No. 2. 

In Figure 6 we have similar parallelograms of twelve sub- 
divisions, six obliquities. 

In Figure 7 we have parallelograms similar in area, but 
with 24 subdivisions, 12 obliquities. 



PLATE NO. 3. 63 

In Figure 8 we have again parallelograms of the same area 
as Figure 1, with 30 subdivisions, 15 obliquities. 

It must also be conceded that the principle thus exemplified 
or demonstrated is capable of any extension ; as you extend a 
parallelogram or triangle, half parallelogram, obliquely be- 
tween the same parallels, precisely as you increase the length, 
you proportionately diminish the breadth. The area will 
always be invariable and the same ; if the extension in length 
were even hundreds, thousands, millions, millions of millions, 
indefinitely without limit, the breadth would be proportionately, 
indefinitely, infinitesimally diminished. 

In Figures No. 9, 10, 11, the same right-angled parallelo- 
gram as in No. 1 is again drawn. The added triangles on 
the right represent various small and fractional degrees of 
obliquity, intermediate between No. 1, the right-angled paral- 
lelogram, and No. 2. But as long as the added triangle on 
the right, however small or fractional, or varying, is exactly 
equal to that which is cut off on the left, the area is absolutely 
the same, and these are only additional examples of the same 
truth. 

Figure 12 exemplifies that what is shown in the preceding 
figures to be true of parallelograms, is also true of half paral- 
lelograms or triangles. Figure 12 is an aggregate of triangles. 
The triangle on the extreme left, and that on the extreme 
right form together the right-angled parallelogram, Figure 1. 
The two next to these from the right and the left form the 
longest diagonal of the oblique parallelogram, No. 2. The 
two next from the right and the left form the oblique parallelo- 
gram, No. 3^ The two next that of No. 4. The two next in 
like manner form No. 5 ; and thus the two triangles of cor- 
responding obliquity represent, and their area is demonstrated 
by viewing them as combined into parallelograms. Thus the 
two central triangles correspond to the oblique parallelogram, 
Figure 8. 



64 EXPLANATION OP THE PLATES. 

The parallelograms are all equal to each other, and contain 
the area of two square inches. The triangles, consequently, 
are all equal, and contain one square inch. 

Figure 13 is another example of the same truth, as is 
shown in Figure 12. If we call from left to right breadth, 
the 32 triangles forming No. 12, are one inch broad, and two 
inches long; while the 16 triangles forming No. 13, are each 
two inches broad, and one inch long. The triangles in Fig- 
ure 13 are of a different form, but equal in area, each being 
one square inch. 

All the figures show that length and breadth justly consid- 
ered are at right-angles with each other, and that area is the 
product of length multiplied by breadth. 



PLATE No. 4. 

This diagram consists of plane figures composed of several 
triangles. In Figure 1, starting at the centre, we have three 
similar and equal angles of 120 degrees, forming three 
triangles, which together form one triangle of equal sides, an 
equilateral triangle. If we take separately one of these three 
triangles, and (as in Plate 2, Figure 12,) double its length and 
breadth, we quadruple its area. If we do the same to each 
of the other two triangles, we in like manner quadruple their 
area. If we now consider the triple triangle as one undivided 
whole, and double its length and breadth, we quadruple its 
area. What is true of each part separately, is true of the 
whole. 

In Figure 2, we have at the centre four angles of 90 
degrees, an aggregate of four triangles forming the well 
known figure called a square. If we double the length and 
breadth of each of these triangles separately, we obviously 
quadruple the area of each, and consequently that of the 



PLATE NO. 4. 65 

square they compose. The side of the inner square is equal 
to the base of one of the triangles, that of the outer square to 
the base of two. Having doubled the length and breadth, we 
quadruple the area. 

In Figure 3 we have five triangles forming the regular 
pentagon or five cornered figure, illustrating the same princi- 
ple. So in Figure 4 we have six triangles forming a six 
cornered figure, a hexagon ; in Figure 5 eight similar triangles 
forming an eight cornered figure, an octagon ; and in Figure 
6 ten similar triangles forming the ten cornered figure, the 
decagon ; each of these illustrating the same principle. 

In Figure 7 we have on the left six triangles forming half 
of the twelve cornered figure, the dodecagon, and on the right 
twelve triangles forming half of the twenty-four cornered 
figure. In Figure 8 we have on the left eight triangles 
forming half of the sixteen cornered figure, and on the right 
sixteen triangles forming half of the thirty- two cornered 
figure. All of these illustrate the same principle. The increas- 
ing number and acuteness of the triangles does not alter the 
rule ; it applies to every triangle, however acute, and to every 
aggregate of triangles, however numerous. 

These aggregates of triangles are frequently called by the 
general name of polygons, that is, many cornered figures. 
In Figure 1 they form a triangle within a circle. Here the 
segments, or interstices between the triangle and the circle, 
are large. As you increase the number of sides, as in 
Figures 2, 3, 4, 5, 6, 7, 8, the segments diminish, and the 
polygons increase, and make an approach towards the circle 
in a manner which can be determined and explained both by 
geometry and arithmetic. 

This approximation towards the circle is capable of being 
continued, the interstices continually diminishing, and becom- 
ing less and less as you increase the number of sides of the 

i 



66 EXPLANATION OP THE PLATES. 

polygon ; but as there must always be a difference, an inter- 
stice, between the straight sides of the polygon and the curve 
line bounding the circle, the approximation can never be 
exhausted. 

This subject, the approach of the polygon toward the circle 
will be farther illustrated in the explanation of Plate No. 5. 



PLATE No. 5. 

In Plate No. 4 we presented examples of a number of 
polygons drawn within the circle, and stated that the relation 
of their areas to each other, and to the circle enclosing them, 
(the circumscribing circle,) was capable of an arithmetical 
solution. 

The relation of multitudes and magnitudes to each other 
are, perhaps, in every case capable of an arithmetical solu- 
tion. [Multitude is but magnitude expressed segregately. 
Magnitude is but multitude expressed aggregately.] A com- 
prehensive knowledge of mathematics includes and compre- 
hends all these relations. It also includes all and every thing 
which can be made the object of algebraic formulas, for 
algebra is either arithmetic or geometry expressed in short 
hand. 

In the notice of Plates No. 1 and 2, we asserted that every 
measurement of two dimensions was really, or was repre- 
sented by length and breadth — length multiplied by breadth, 
and that this law was invariable, and applicable to every 
form. 

In Plate No. 5, we have a square. It is intended to be a 
square foot, but our remarks will apply equally to any square, 
without regard to size. It is divided by straight lines inter- 
secting at the centre, into four equal squares. Each of these 
squares is again divided diagonally into two equal triangles. 



PLATE NO. 5. 67 

The four central triangles form another square of half the 
area of the first mentioned larger square, which comprises 
eight triangles, or four squares, as at first divided. 

From the centre let a circle now be drawn, whose radius 
is six inches, whose diameter is one foot. The larger square 
outside comprising this circle is called the circumscribing 
square ; the inside square is called the inscribed square. The 
last is obviously half of the first. But the purpose of this 
diagram as well as of No. 6, is to determine the area of the 
circle, and its relation both to the area of the inscribed, and 
to that of the circumscribed squares. 

No other method has been, and it is believed no other 
method ever will or can be devised than that invented twenty- 
one centuries ago by Archimedes, and which he called the 
Method of Exhaustions. It is dependent upon the truth of the 
problem of Pythagoras, the 47th of the 1st book of Euclid's 
Elements, and will be more completely understood on revision, 
after a full comprehension of that problem, which is illustrated 
in the 8th, 11th, 12th, 13th, and several subsequent Plates of 
this work. 

We have repeatedly stated that length and breadth is the 
invariable law of all superficial figures ; and this is indisputably 
true. But while all figures bounded by straight lines are 
capable of being compared with, and measured by each other 
with facility, all figures bounded by curve lines, as the circle, 
and all varieties of the ellipse, can only be determined by 
approximation. The method of Exhaustions of Archimedes is 
an example of this. 

To determine the area of the circle to its circumscribed 
square. If the radius be 1, the diameter will be 2, and the 
area of the circumscribed square, that is the square of the 
diameter, will be 4, and the area of the inscribed square will 
be exactly half of the area of the inscribed square. 



68 EXPLANATION OF THE PLATES. 

The areas of the circumscribed and inscribed squares 
being thus accurately measured, in order to determine the 
area of the circle it is simply necessary to calculate the ratio 
between these squares and the circumscribed and inscribed 
polygons of double the number of sides, that is eight sides. 
Then the same ratio for polygons of sixteen sides, then of 
thirty-two sides, &c. &c, until the series be carried to an 
infinite extent. Half of the sum of the differences of the two 
polygons will give an approximate area, infinitely close. 

If, as in Plate 5, above, on the left, we first divide the side 
of the smaller or inscribed square into two equal parts, and, 
secondly, draw a line from the centre to the circumference of 
the circle, through the dividing point, and then draw lines from 
each corner of the square, meeting and touching the circle at 
this line, we shall form the octagon. A line which touches a 
circle is called a tangent. 

As the triangles thus cut out of the segment of the circle 
and added to the square in forming the octagon are right- 
angled, and as their length and breadth can be geometrically 
determined, their area is of course demonstrable. 

Again, at the top of the figure proceeding to the right, we 
form in a similar manner from the octagon, the sixteen sided 
figure, inclosing other right-angled triangles, whose length, 
breadth, and area are demonstrable. 

On the right of the Plate, from the sixteen sided figure we 
form the thirty-two sided figure, including other right-angled 
triangles, whose length, breadth, and consequent area are also 
capable of demonstration. 

And thus again, from the thirty-two sided figure we form 
the sixty- four sided figure, which is as : far as can be drawn 
distinctly on so small a scale as this diagram. It is obvious 
that a straight line and a curve line must, if placed side by 
side, always have an interstice between them, and therefore 



PLATE NO. 5. 69 

the straight sides of any polygon cannot ever be exactly com- 
mensurate or identical with the curve line of the circle, 
though these will continually approximate, and by a rapid 
ratio ; and though we may proceed by decimals, arithmeti- 
cally, and make approximations without limit in number, they 
can never exactly coincide with the circle. 

For a familiar illustration of the possibility of a continued 
approximation without limit, I would present to each student 
ten thousand dollars, on the condition that he should make 
twenty successive divisions of the amount, halve it, and again 
halve it, and return the halves to me. 

Thus, 1st, 10,000 divided by 2 = 5,000. 2d, 5,000 by 2 
= 2,500. 3d, 2,500 by 2 = 1,250. 4th, 1,250 by 2 = 625. 
5th, 625 by 2 = 312.50. 6th, 312.50 by 2 = 156ft. 7th, 
156$, by 2 = 78ft- 8th, 78ft by 2 = 39^. 9th, 39ft by 
2 = 10ft. 10th, 19ft by 2 = 9ft. 11th, 9ft by 2 = 4ft. 
12th, 4ft by 2 = 2ft. 13th, 2ft by 2 = 1ft. 14th, 1ft by 
2 = ft. 15th, 61 cents by 2 = 30J. 16th, 30| by 2 = 15 J. 
17th, 15J by 2 = 7|. 18th, 7f by 2 = 3Jf. 19th, 3i§ by 2 
= lif, nearly. 20th, 1£§ by 2 = |J of a cent. 

We have thus in twenty successive divisions cut down the 
amount to less than one cent, or less than one millionth of the 
original amount. $10,000 X 100 cts. == 1,000,000; but we 
might by adding six decimals again make twenty successive 
divisions, reducing the amount to one millionth of one mil- 
lionth, and again, by adding decimals, to an indefinite extent. 
But as our plan prescribes a continued division, the remainder 
is never exhausted. So that in theory, and in truth, the entire 
sum of ten thousand dollars would never be returned ; there 
would always be a fractional remainder left. 

The cent being the smallest in value of any coin in use, 
and the ^condition attached to my apparently very generous 
proposal, having diminished the gift to an amount less than 



70 EXPLANATION Of THE PLATES. 

one cent, it amounts practically to nothing at all. "I don't 
owe you a cent." 

A very good illustration, though not so obvious a one, is 
afforded by the air pump. 

The application of this example of continued division ap- 
plies beautifully, and with increased force to the geometrical 
exhaustion of Plates No. 5 and 6, as the portions between the 
inscribed square and the circle, and those between the circle 
and the circumscribing square, (the portions included and ex- 
cluded, the integral and differential areas,) amount to more 
than one half; the ratio of approximation exceeds three quar- 
ters, the effect of which is to exhaust the integral and 
differential to less than one millionth in ten approximations, 
instead of twenty. 

Say 1st, to {. 2d, \ of { = £. 3d, \ of & = &. 4th \ of 

A = ^ 5th > i of nh = lfa- 6th > i of tf&t = *Ai- 7th > 
i of iws ** tmoo» nearly. 8th, | of ^ ■ = ^ m . 9th, J of 
nk = sireW 10tn > i i = loooooo - ^ r > ™ ore accurately, 
suppose the radius of a circle to be denoted by unit or 1 ; 
the area of the circumscribing square will be expressed by 4, 
and the area of the inscribed square by 2. 

Wherefore, the surface of the inscribed octagon is = */2 
X 4 = 2 .8284271, and the surface of the circumscribing 
octagon is found by the analogy 2+2 .8284278 : 2X2 : : 4 : 
3 .3137085. Again, 7(2 .8284271 X 3 .3137085)=3 .0614674, 
which expresses the area of the inscribed polygon of 16 sides, 
and 2 .8284271+ 3 .0614674 : 2X2 .8284271, or 5 .8898945 
: 5.6568542 : : 3 .3137085 : 3 .1825979, which denotes the 
area of the circumscribing polygon of 16 sides. Pursuing 
this mode of calculation, by alternately extracting a square 
root, and finding a fourth proportional, the following Table 
will be formed, in which the numbers expressing the surfaces 
of the inscribed and circumscribed polygons continually 



PLATE NO. 6. 



71 



approach each other, and consequently, to the measure of 
their intermediate circle. 



ER OP SIDES. 


AREA OF INSCRIBED 


AREA OP CIRCUMSCRIBING 




POLYGON. 


POLYGON. 


4 


2 .0000000 


4 .0000000 


8 


2 .8284271 


3 .3137085 


16 


3 .0614674 


3 .1825979 


32 


3 .1214451 


3 .1517249 


64 


3 .1365485 


3.1441184 


128 


3 .1403311 


3 .1422236 


256 


3 .1412772 


3 .1417504 


512 


3 .1415138 


3 .1416321 


1024 


3 .1415729 


3 .1416025 


2048 


3 .1415877 


3 .1415951 


4096 


3 .1415914 


3 .1415933 


8192 


3 .1415923 


3 .1415928 


16384 


3 .1415925 


3 .1415927 


32768 


3 .1415926 


3 .1415926 



What is called squaring the circle, is simply ascertaining 
the area of the circle relatively to that of the square. 



PLATE No. 6. 

Is a copy of Plate No. 5, with the addition that in Plate 5 
the polygons based on the inscribed square only are exempli- 
fied. In No. 6, the polygons based on, and measured from 
the circumscribed square are added. The usual name for 
this process is the integral and differential calculus. It is used 
in approximating all curve lines, as the ellipses of every 
sort. A circle is a square cornered off. An ellipse is an 
oblong cornered off in a similar or analogous manner. 

We have thus endeavored to give an idea of the relation 
which the area of the circle bears to that of the square. 



72 



EXPLANATION OF THE PLATES. 



PLATE No. 7. 

We have more than once stated as an axiom, that the 
areas of all plane figures, of whatever form, whether bounded 
by straight lines or curved lines, are results of the length 
added to, or multiplied by the breadth. 

The purpose of this Plate is to show that the relation of the 
area of any circle to the square including it, the circumscribed 
square, is an uniform one, whatever may be its size. 

In this Plate, Figure 1, we have a square inch, within 
which is described a circle, and within the circle an octagon. 
The circle is the square cornered down to an uniform curve 
returning into itself. The area of the circle we have shown 
in Plates No. 5 and No. 6, is nearly as .7854, the square being 
10 .000 or 31416, the square being 40 .000. For the present 
purpose it is only necessary to make self-evident that the 
relation of any circle to the square circumscribing it is inva- 
riably similar. 

In Figure 2 we have doubled the length and breadth of the 
square, and consequently the length and breadth of the 
inscribed circle and octagon ; and have quadrupled their area. 
In Figure 3 we again double the length and breadth of Fig- 
ure 2. In Figure 4 we double the length and breadth of 
Figure 3. In each case we have quadrupled the area. We 
have thus in Figure 3, 4 inches X 4 inches =16 inches area, 
and in Figure 4, 8 X 8 = 64 inches area, or 64 times the area 
of Figure 1 ; and the corresponding circle and octagon are 
64 times the area of those in Figure 1. 

In Figure 5, for further illustration, we have combined 
Figures 1, 2, 3, 4. It is obvious that the areas of each and 
all the constituent parts of these figures bear an uniform ratio 
to each other ; not only the squares, octagons, triangles, and 
circles, but also the integral segments between the octagons 
and the circles, and the differential portions which form the 



PLATE NO. 7. 73 

corners between the circles and the squares. The length 
and breadth of the square being successively doubled, and the 
area quadrupled, all of the constituent parts which compose 
the square are increased in the same proportion — in the same 
ratio. The usual axiom is, all circles are to each other as 
the squares of their diameters. All plane figures are to each 
other as the squares of their homologous sides. [Homologous 
is a Greek word signifying similarly described.] But there 
would be equal truth and propriety in saying or teaching as 
an axiom, all squares are to each other as the circles of their 
diameters. All plane figures are to each other as the circles 
of their homologous sides. The real truth is that all plane 
figures, whatever may be their outline, regular or irregular, 
bounded by straight lines or bounded by curve lines, whether 
squares, octagons, or triangles, whether circles or ovals, or 
any irregular figure, are to each other as their length multi- 
plied by their breadth. 

In Figure 6 we have at the centre the circle of one inch in 
diameter, and successively circles of two, three, four, five, six, 
seven, and eight inches in diameter. These diameters of 
course represent length and breadth, and multiplied, each into 
itself, give areas of one, four, nine, sixteen, twenty-five, thirty- 
six, forty-nine, sixty-four, for the areas of the respective 
circles. 



PLATE No. 8. 

The fine geometrical lines on which this figure is con- 
structed, were drawn originally with all practicable accuracy. 
The stronger black and red lines are drawn within, and 
entirely independent of these, and are solely for a more distinct 
impression on the eye. The coloring which I have thought 
it proper to use must be regarded as an expedient independent 



74 EXPLANATION OF THE PLATES. 

of the geometrical value of the diagram, and entirely defensi- 
ble only on the plea of greater distinctness. 

Drawn in black from the centre, we have the square inch, 
and the circle of two inches diameter, whose radius is the 
side of this square. Above, and parallel with this on the left, 
we have the square of two inches sides and its corresponding 
circle. On the right, occupying a similar relation to the 
centre, we have the square of four inches sides, and that of 
eight inches sides, with their respective circles. 

Drawn in red, from the centre on the diagonal of the first 
square inch, we form the second square, which (by the great 
problem of Pythagoras, the 47th of the first book of Euclid,) 
doubles the area of the square inch. It is our first demonstra- 
tion, and probably the simplest which can possibly be given of 
this invariable problem, that the square drawn upon the hypo- 
thenuse of any right-angled triangle is equal to the sum of the 
squares {which may be drawn) on the other two sides. The 
squares of the two sides are faintly drawn, and each side being 
one inch, these squares are square inches, and of course equal 
to the original square inch. If we now consider square No. 2 
as divided diagonally into four equal triangles, one of these 
triangles will be the half of the square No. 1, and square No. 2 
therefore obviously doubles the area of square No. 1. 

Parallel with square No. 2, the square No. 4, and the 
square No. 6, both drawn in red from the centre on the 
diagonal of squares No. 3 and No. 5, repeat the same problem. 
We have therefore the four black squares, (with their respect- 
ive circles,) the relation of which, each to the other, we can 
demonstrate by drawing two lines parallel with the sides of 
the larger square, intersecting it at its centre, dividing it into 
four squares, each equal in area to the smaller. And also the 
three red squares with their respective circles, the relation of 
which to each other can be demonstrated in the same manner, 



PLATE NO. 8. 75 

by drawing two lines parallel with the sides of the larger 
square, intersecting at its centre. While at the same time the 
relation of the black squares to the red squares is alternately 
and successively capable of demonstration by diagonal divi- 
sion into four triangles, intersecting at the centre of the square, 
forming examples of the problem of Pythagoras. 

The relation of the squares and the circles being in every 
case similar, they respectively double each other's areas as 
you build up from the square inch, and diminish by a similar 
ratio in reversing the process. 

We may start at the centre, that is at the one square inch, 
or the triangle, which is one half of this square, and may 
demonstrate the area synthetically, by addition, or we may 
start from the largest square of eight inches sides, and pro- 
ceed by division, analytically, towards the smaller areas. 
There is no difference in principle, synthesis is but addition or 
multiplication, analysis is but subtraction or division. 

They double each other's areas successively, the ratio of 
the squares and circles remaining unchanged. 

Whenever a figure is bounded by straight lines, its area is 
capable of most prompt and definite measurement. Where a 
figure is bounded by curve lines, its area, compared with fig- 
ures bounded by straight lines must be ascertained by approxi- 
mation. By using a curve line figure for our unit of mea- 
surement, as a circle of one inch in diameter, we can measure 
similar curved line figures with greater simplicity ; thus, we 
can definitely measure the relation of circles to each other by 
simply multiplying their diameters, that is, their length and 
breadth; but comparing them with squares or right lined 
figures is another thing, and this must be done by approxima- 
tion. 

THE APPLICATION OR USE OF PLATE NO. 8. 

The distinction between pure and applied mathematics as 



?6 



EXPLANATION OF THE PLATES. 



usually made is, in our judgment, vague, and thus is produc- 
tive of great uncertainty to the learner, unless especial care 
is taken to make a true definition of both. And if this is done, 
there will be found to be in principle no difference whatever, 
and none, in fact, but what is inseparable from the imperfect 
application of human measurements and human mechanism. 
It is true, that in applied mathematics we often have to take 
into consideration some physical fact, inaccuracy, impediment, 
or source of error, which alters the exact mathematical result ; 
but in this case we always have reason to attribute the error 
to some oversight, disturbance, or irregularity resulting from 
our own incomplete comprehension, or from imperfect appli- 
cation, and not to any defect in the principle itself. 

In Euclid we have first, the book called Geometry, and 
secondly Trigonometry, and subsequently ten other books, in 
which proportion, &c. are treated. The successors of Euclid 
have added three more books, so that we have fifteen books 
of " Pure Mathematics." What is pure mathematics ? Let 
us consider. 

The instant we begin to make any application of mathe- 
matical principles to mechanics or to natural philosophy, we 
are told that is not pure mathematics. Then what is pure 
mathematics? Pure mathematics, we are given to under- 
stand, is Geometry, and Trigonometry, and the whole twelve 
books of Euclid. But the figures drawn on the paper in 
exemplification of the problems are not pure mathematics, for 
they are subject to the inaccuracy of drawing. If you say 
that two and two make four, this is pure mathematics. If 
you say that ten yards and ten yards are twenty yards, or 
that ten yards in length and ten yards in breadth are one 
hundred square yards in area, these are pure mathematics. 

But if you come to make the actual measurement, you are 
subject to some inaccuracy, which, as already stated, is in- 



PLATE NO. 8. 77 

separable from all human measurements, and human mechan- 
ism. In mechanics you have some friction. In astronomical 
observations you are subject to more or less mist in the atmo- 
sphere, and you have to make allowance for refraction. 

The necessary allowance for these being made, pure 
mathematics and mixed mathematics assimilate ; they are the 
same thing. Pure mathematics is the mathematical truth as 
it exists in your mind. 

The perfect intellectual conception of a 'law of Nature is 
pure mathematics. But in Nature the combined action of 
various laws modifies the application of such a law in particu- 
lar instances, and this must be allowed for. 

In applied mathematics, therefore, we approach to absolute 
truth as nearly as we can ; but we have to make allowance 
for our own errors of observation, for imperfection of our 
instruments, and for disturbing or contravening causes in 
Nature ; and, therefore, though we know that we are correct, 
and can prove it, we always admit the universally acknow- 
ledged liability to some inaccuracy. It is thus that the most 
skilful accountant never omits to write " errors excepted." 

It is in astronomy, however, that applied mathematics 
approaches most nearly to absolute truth. It is here that the 
accurate accordance of our measurements of the revolutions 
of our sister planets, with those diurnal and annual revolutions 
of the orb we inhabit, by which we measure terrestrial time, 
and witness the sure return of the seasons recorded among 
the stars : enables us to compare them, and to predict with 
a precision almost identical with absolute truth the transit of 
a planet, and the occurrence of eclipses, over long intervening 
centuries. 

We come now to consider Plate No. 8 as a contrivance for 
illustrating, explaining, or demonstrating mathematical princi- 
ples. We will first apply it to mechanics, and secondly to 
natural philosophy. 



78 EXPLANATION OF THE PLATES. 

Let us now consider Plate No. 8, as representing a wheel. 
The vertical line from the centre, eight inches in length, may 
represent a radius, a spoke, a lever of eight inches, for a 
wheel is an aggregate of levers. 

If we apply a force at the end of the lever, the centre being 
the fulcrum, we necessarily exert our force in the circumfer- 
ence of the largest circle, say to the right. If our power is 
applied at the circumference of the smallest circle, one inch 
beyond the fulcrum, it will be exerted through one eighth of 
the distance, with one eighth of the velocity, but with eight 
times the power. 

If our power is applied two inches beyond the fulcrum, it 
will be exerted in the circumference of the third circle of two 
inches radius, through one quarter the distance, with one 
quarter the velocity,, but with four times the power. 

If our power is applied four inches beyond the fulcrum, it 
will be exerted in the circumference of the fifth circle of four 
inches radius, through one half the distance, with one half the 
velocity, but with twice the power of the force exerted. 

If our lever or radius were extended to the same length on 
the opposite side, we should apply our power in the circum- 
ference of the same largest circle, and of course through the 
same distance, with the same velocity, and with the same 
power. It would be a simple balance. 

It is obvious, that if instead of the longer end of the lever, 
we were to apply the force at the other or shorter end of the 
lever, we should exert our power in the circumference of the 
larger circles with increased velocity, and through increased 
distances, but with proportionally diminished power. It is an 
exemplification of the truth that what is gained in time is lost 
in power, and the reverse. 

This diagram also well exemplifies what is called in the 
books the second form of the lever. In this case, the lever 



PLATE NO. 8. 79 

terminates at the fulcrum, that is, at the centre. It is, conse- 
quently, only on our own side of the fulcrum that such a lever 
can act, and the direction of the power is the same, or rather 
in a curvilinear 'parallel with the force applied. 

It also exemplifies the third form of the lever as it is called, 
where the force is applied between the fulcrum and the resist- 
ance. This is, in truth, only the reverse of the second form 
of the lever. For illustration: Let the black line of eight 
inches represent a ladder of a given length, lying on the 
ground; let the ground at the centre be the fulcrum. If a 
force is now applied at one inch from the centre, and exerted 
through a given distance, in the circumference of the first 
circle, it will raise the top of the ladder through eight times 
this distance. But it will require eight times the force. Eight 
times the force exerted through one distance, is thus equiva- 
lent to one force exerted through eight distances. 

The diagram also well illustrates the bent lever, or the 
crank motion. The bent lever does not alter the principle. 
It is obviously not different in principle from the straight lever. 
Bending it very little or bending it very much makes no differ- 
ence. Its relative force is measured by the distance from the 
centre, or fulcrum, in either and every case. 

The diagram also exemplifies the wheel and axle, the capstan 
and the pulley ; the w r heel is but an aggregate of levers, a 
perpetual lever. If we apply our force in the circumference 
of the largest circle, and wind up our rope upon the smallest 
circle, we shall wind up the rope through only one eighth of 
the distance in which the force is exerted, but we shall wind 
it up with eight times the power. 

There is no form of the lever which this diagram does not 
represent. Thus much for mechanics. 

Plate No. 8 also well exemplifies the balance and the 
steelyard. 



80 EXPLANATION OF THE PLATES. 

Let the black line vertical from the centre, eight inches in 
length be one arm of a balance, and let us suppose the oppo- 
site arm to be extended equally on the opposite side. Let the 
centre be the fulcrum, and let the right side of the plate be 
the top. One weight, say one pound weight, at eight inches 
from the fulcrum balances a similar weight at a similar dis- 
tance opposite ; and the vibration of the weights on either side 
will be through equal distances, and in the circumference of 
the same largest circle. It is obvious that this would repre- 
sent the simplest form of the balance. Any similar weights 
at similar distances will of course balance each other. 

But the one pound weight at eight inches distance may also 
be balanced by two pounds at four inches from the fulcrum, 
vibrating in the circumference of the circle of four inches 
radius, through one half the distance described by the one 
pound. 

The one pound at eight inches may also be balanced by 
four pounds at two inches distance from the fulcrum, vibrating 
in the circumference of the circle of two inches radius, 
through one quarter of the distance described by the one 
pound. 

And the one pound at eight inches may again be balanced 
by eight pounds at one inch from the fulcrum, vibrating in the 
circumference of the circle of one inch radius, through one 
eighth of the distance described by the one pound at eight 
inches radius. It is plain that this is a simple linear measure- 
ment, and that distance and weight represent each other. 

Thus : One pound weight, or any one weight at eight 
distances = 8 

Eight weights, say eight pounds weight, at one 
distance = 8 

Also : Two weights, say two tons, at four distances, =s 8 

Four tons at two distances, = 8 



PLATE NO. 8. 81 

There might be any other representatives of distance and 
of weight which, multiplied into each other would would give 
the result, 8, which of course would interchangeably balance 
with any of these, and with each other. 

Sixteen pounds weight, at one half a distance, - - = 8 
One half of one pound weight, at sixteen distances, - = 8 
Thirty-two pounds, at one quarter of a distance, - = 8 

One quarter of one pound, at thirty-two distances, - = 8 
Also VS pounds at >/8 distances - - - - = 8 
Plate No. 8 also well illustrates Natural Philosophy. What 
is Natural Philosophy ? It is the laws of certain natural phe- 
nomena, the most remarkable of which are heat, light, sound, 
magnetism, electricity, galvanism, gravitation. 

Let our first illustration be of light. Let the centre of the 
diagram be the pupil of the eye. If an object measured, let 
us suppose from the vertical line to the right, at the distance 
of one inch from the centre, occupy one eighth of the circum- 
ference, or 45 degrees of the smallest circle, the same object, 
at eight times the distance, at the circumference of the 
largest circle, would occupy only an eighth of one eighth, or 
five and five eighth degrees of this circumference. This is 
perspective. This illustration is on a small scale ; but if 
instead of one inch we commence with a foot, or a yard, or a 
mile, the principle would be the same. 

In further exemplification, let us suppose ourselves on the 
summit of a very high mountain on a fine day ; the extent of 
our vision might be such, that the one inch might represent 10 
miles, the two inches 20 miles, the four inches 40 miles, the 
eight inches 80 miles; every object, from the bush at our feet 
in the immediate foreground, to the most distant object in the 
extended landscape, would occupy in our field of vision its 
true mathematical relation to other objects, according to its 
distance in terrestrial perspective. 

K 



82 EXPLANATION OF THE PLATES. 

If we extend our vision beyond the earth — to the heavens, 
the same principle will be found to govern celestial perspect- 
ive. Parallax is but another word for the same thing. Par- 
allax is but the perspective of the heavens. We divide our 
circumference into 360 equal parts, called degrees. Thus 
when we say the moon's diameter is a little more than half of 
a degree, we mean that it would take about 180 moon's 
diameters to reach from the zenith to the horizon, or about 
four times this number, 720, to form a continuous ring of 
moons encircling the whole horizon. 

Now it is obvious, that if the moon were nearer to us, say 
only one eighth of its present distance, it would require only 
one eighth of the number of moons to form the entire ring. 
The moon would then have eight times its present apparent 
diameter, and the earth would receive from the moon 64 
times as much of the moon's reflected light as it does at 
present. 

For the more accurate measurements, including those of 
the celestial bodies and their motions, the degree is divided 
into 60 parts called minutes, and the minutes are again sub- 
divided into 60 parts, called seconds. 60 X 60 = 3600, 
there are 3600 seconds in each degree, and 360 degrees in 
the circumference. 3600 X 360 gives 1,296,000 seconds. 
When we speak of a second, we mean the m£<ro?r part of a 
circumference. 

With the present excellence of our instruments and power 
of comparing our measurements of celestial distances with 
accurate measurements of time, the science of astronomy is 
justly regarded as the most remarkable result of the human 
intellect. 

Plate No. 8 is available for other illustrations. We have 
considered light as a radiating phenomenon, as a radiating 
force. Heat also radiates by a similar or analogous law. 



PLATE NO. 9. 83 

Magnetism, Electricity, Sound, Gravitation, have similar or 
analogous relations to the laws of measurement. All radiating 
phenomena illustrate and tend to explain each other. 

In illustrating the radiation of heat, let the smallest circle in 
Plate 8 be a stove, or the half of this circle be a grate, of a 
cold day. If four persons at one distance (represented by the 
one inch,) each occupy in the curve of the five side circle 45 
degrees, the same persons at two inches distance would 
occupy only half of this portion ; there would be room for 
eight at twice the distance, for sixteen at four times the dis- 
tance, and for thirty-two at eight times the distance. The 
proportion of radiant heat will of course be diminished to the 
same extent in which the persons of the front rank eclipse or 
intercept the radiant heat. It is also obvious that this is not a 
simple linear relation. The persons of the front rank will 
intercept twice as much breadth of radiation as those of the 
second, four times as much as those of the third, and eight 
times as much as those of the fourth, so that here is a consid- 
eration, a measurement of both length and breadth. It be- 
longs to the second class of measurements. Light and heat 
radiating in all directions, are diffused as from a centre, on 
all sides, and belong to the third class of measurements. But 
light and heat received upon a plane surface is a measurement 
of the second class. It is as the length and breadth of the 
given surface. " It is as the squares of the distances. , ' One 
person in the front rank would thus intercept the radiant heat 
from 64 persons occupying similar space at eight times the 
distance. 



PLATE No. 9. 
Plate No. 9, although at first sight very different from ~No. 
8, is in principle the same. 



84 EXPLANATION OF THE PLATES. 

In No. 8 the squares constitute an essential part of the 
figure. It is on the sides of the squares that the circles are 
drawn. No. 9 is not different in principle or in construction, 
only that the squares are omitted. 

In drawing the original of No. 9, the squares were first 
drawn in pencil with great care. The circles were then 
drawn, and the squares erased. The circles are therefore to 
be relied on, as exactly the same as those of No. 8. 

Starting from the central, or smallest circle of one inch 
radius, the circles double each other in area as we proceed 
outward toward the outer or larger circle, as is shown in the 
Plate 8. An ellipse is drawn, enclosing, and almost touching 
the smallest circle, and cutting each of the others. In pur- 
suance of our purpose to illustrate radiant forces, let us first 
suppose this ellipse to represent the walls of an oval room, and 
let the first illustration be of light. 

Let a common lamp be placed at the centre of the circles, 
which is one of the foci of the ellipse. The light radiating 
from this point would first meet the ellipse at the distance of 
(at a radius of) one inch, and its diffusion would be repre- 
sented by the area of the smallest circle. The light reaching 
the ellipse at the intersection of the third, of the fifth, and of 
the seventh circles, is at radii of two, of four, and of eight 
inches respectively, and its diffusion is as the areas of these 
circles, that is, as the length multiplied by the breadth. The 
radius of the first circle 1X1 = 1. 2x2 = 4. 4x4 
= 16, and 8 X 8 = 64, the area of the largest circle. 

The old expression is, light diffuses itself as the squares of 
the distances. But it must be borne in mind that as light dif- 
fuses itself, its intensity diminishes. The intensity of light is 
inversely as the squares of the distance. How is this ? Let 
us consider. Light does not diffuse itself in squares. Light 
radiates from a centre, and the radii may be very naturally 



PLATE NO. 9. 85 

represented by circles. It is true, that these circles are to 
each other as the squares which might be drawn on their radii. 
It would be equally just to say that light diffuses as the octa- 
gons of the distance, or as the triangles or hexagons of the 
distance. It would be more intelligible, perhaps, than either, 
to say that light varies in intensity, diminishing as the area of 
diffusion increases, that is, as the length and breadth of the 
radii multiplied into each other. Its diffusion is obviously not 
a mere line. It has length and breadth. It belongs, at least, 
to the second class of measurements. 

Light is a radiant force. Heat also is a radiant force. 
The same law applies to all radiant forces. Magnetism, 
Electricity, Sound, Gravity are all radiant phenomena. 

Sound acts — reverberates — from its source or centre, in a 
manner which is capable of the same measurement. In ap- 
proaching the ear of our audience, we gain in an analogous 
manner the advantage of the squares of the distance. 

We may also well exemplify the mechanical construction 
of the wheel and the lever by this diagram. Let the ellipse 
be an eccentric wheel, for instance, a cam in a steam engine, 
its axis being the centre of the circles. A wheel is an aggre- 
gate of levers. The shortest radius of our eccentric wheel is 
a lever of one inch. The longest radius is a lever of nine 
inches. Between these we have levers of all intermediate 
lengths. 

The eccentric wheel is thus an aggregate of levers. It 
may be practically considered as a short lever and a long 
lever gracefully curved, so as to pass most easily from one 
into the other. 

This diagram also illustrates perspective. If an object 
occupy in the field of vision at the distance represented by the 
largest circle, say the breadth of the portion which is colored, 
or 5| degrees, it will at the distances of the nearer ciucles, at 



86 EXPLANATION OF THE PLATES. 

the 6th, 5th, 4th, 3d, and 2d circles, occupy proportional 
breadths of llj, 22^, 45, and 90 degrees, represented by the 
differently colored portions of these circles respectively. 

But perhaps the most interesting application of Plate No. 9 
is the exemplification of Gravity. Light and heat are exam- 
ples of radiant forces whose action is centrifugal. Gravity is 
an example of a radiant force, whose action is centripetal. 
Light and heat are diffused from a centre by a law which we 
have exemplified. Gravity is attractive towards a centre. 
But the relations of both, and of all to the laws of measure- 
ment, beautifully illustrate and explain each other. 

We will here transcribe or repeat our concluding remarks 
on the explanation of Plate No. 1. "The most important 
primary lesson which can be taught to the student of mathe- 
matics is the natural and direct connection of the three classes 
of measurements, with all the varieties and all the relations of 
motion. 

Every motion is a result of force. The nature of force is 
not easily understood, but we can observe and measure the 
motion which is its result. One force produces motion in a 
straight line, and only in a straight line. Every motion which 
is in a straight line is a result of one single force, (or it may 
be of one aggregate force.) It represents, and is capable of 
being measured by the first class of measurements. 

A motion which is not in a straight line must always be a 
result of more than one force. Thus, a stone thrown into the 
air returns to the earth. It has a curved linear motion. The 
projectile force would have impelled it in a straight line, but 
gravity, a second force, acting upon it perpendicularly, brings 
it to the ground. In like manner, the movement of a stone 
attached to a string held in the hand, describes a curved 
linear motion. So also do the Planets in their revolutions 
round the sun, and of their satellites round the Planets." 



PLATE NO. 9. 87 

But to return to Plate No. 9, in illustration of Gravity. If a 
cannon ball were fired, or a stone thrown directly upward 
toward the zenith, it would lose its velocity inversely, as " the 
squares of the distances ;" (that is, as the circles of diffusion 
represented in the Plate, through which it was impelled,) till it 
reached the highest or culminating point ; it would then begin 
to fall, and in common with all falling bodies, would acquire 
velocity as " the squares of the distances " through which it 
fell ; that is, as represented by the same circles in the Plate, 
which we will now call circles of concentration, falling 16 
feet in the first second of time, 48 feet in the second, 80 feet 
in the third second, and so on, and making its descent in the 
precise time in which its ascent was impelled. If the stone or 
cannon ball were thrown obliquely upwards, it would describe 
a curved line, but the same law of gravity would determine 
its ascent and descent. 

If a cannon ball impelled by the greatest practicable force 
were fired horizontally from an elevated position, say from the 
fortress of Gibraltar, and at the same instant another ball were 
quietly dropped from the same elevation, they would fall into 
the sea at the same precise instant of time. The horizontal 
force would have propelled the first several miles, but it would 
have fallen as fast as the other, the action of gravity would 
have been exactly the same on both. The two forces in the 
one horizontally impelled, would not have interfered in the 
slightest degree with each other. 

For a simpler illustration. Let us suppose a ball lying on a 
table to be pushed horizontally, first by a given force in one 
direction, and then, after it was at rest, a second force to act 
upon it horizontally in another direction, say at right-angles 
with the first, its place of rest would be the same as if the two 
forces acting upon it were combined at the same instant to 
impel it. It would have described a path representing the 



88 EXPLANATION OP THE PLATES. 

diagonal of the two forces, and thus have arrived at the same 
point. 

If we suppose one force to represent length, and tfie other 
breadth, the analogy is perfect in every possible variety and 
number of instances, the combined force will, both in its 
intensity and direction, be exactly represented by the diagonal, 
the hypothenuse. 

Further to illustrate what are called centrifugal and cen- 
tripetal forces, let us turn to Plate No. 5, and let us suppose a 
ball at the centre of any one of the four sides of the outer or 
circumscribed square. If a force act on the ball parallel with 
the side of the square sufficient to impel it six inches, it would 
rest at the corner of the square. If a second force now act 
on the ball at right-angles with the first, and in the direction 
of the line of the next side of the square, sufficient to impel it 
six inches it would reach and remain at the centre of the next 
side. But if these two forces acted on the ball at the same 
precise instant, it would take — its path would describe — the 
diagonal line (represented in the Plate by the side of the 
inscribed square,) to the same point. If we suppose one of the 
forces to be an impulse like that of the cannon ball fired hori- 
zontally from a mountain fortress, and the other a constant 
force like gravity, acting slowly at first, but accumulating 
every instant on the rapidly propelled but surely falling body, 
the path described would be, not the diagonal, but the curved 
line of the quarter circle, as in the Plate. This supposes the 
two forces to be equal ; if they were not equal, the path could 
not be a circle. It would then be some other curve, repre- 
senting a portion of an ellipse, or a portion of some other 
section of a cone. 

It obviously requires more than one force, that is, it requires 
two forces to sustain any body revolving around another. 
Gravity is centripetal, and would rapidly attract it to the 



PLATE NO. 9. 89 

centre, the equilibrium of revolution ; the centrifugal force, on 
the contrary, if it were the only one, would carry it off in a 
straight line, as is usually said, in a tangent. 

For a body thus to maintain a circular orbit round another, 
requires two forces, and these must exactly and constantly 
balance each other, describing an uniform curve, returning 
into itself, passing through uniform linear distances, and sweep- 
ing uniform and similar areas in uniform times, and with 
uniform angular velocity. 

To sustain an elliptical orbit of revolution, the two forces 
need not be, cannot be always uniform and equal, as in the 
circular orbit, but even in this case, though the two forces are 
not constant and uniform as in the circle, each of the half 
ellipses measured from the points nearest to, and farthest from 
the centre of revolution, must balance and compensate each 
other, must (as we shall learn,) in every case be equal to, and 
represent each other both in linear distance, in area, and in 
time. 

Let us now suppose the centre of the circles in Plate No. 9 
to represent the sun or centre of attraction, and the circles 
drawn on the Plate to represent the orbits of the Planets. As 
the attraction of gravitation acting upon the nearer, compared 
with the more distant Planets, is found to be measured, not by 
their linear distance from the sun, but to be measured by the 
length multiplied by the breadth of this distance, and is repre- 
sented by the areas of their respective orbits of revolution. 
By the same law, a body, whether a Planet or a Comet, per- 
forming a revolution which is a very little or very much 
elliptical, being acted upon at the nearer (the perihelion) 
portion of its orbit, by a stronger force of gravity, and at the 
further (the aphelion) portion of its path by a diminished force 
of gravity, the centrifugal or projectile force must represent, 
balance, and correspondingly increase and diminish in precise 

L 



90 



EXPLANATION OF THE PLATES. 



analogy to that of the gravity, at every portion of its path, or 
the gravity which might be measured by supposing an unlimi- 
ted number of circles intersecting every point of its orbit. 

The orbits of the celestial bodies, whether Planets or 
Comets, are obviously not straight lines. They are not results 
of one force — they do not belong to the first class of measure- 
ments. They are the result of two forces, the centrifugal 
and the centripetal ; they are represented by planes ; they 
have length and breadth ; they are clearly belonging to the 
second class of measurements. 

For the just conception of the laws of their motions, we are 
indebted to the labors of the immortal Kepler. 

We will here devote a paragraph to the history of Astrono- 
my, previous to the time of Kepler. 

Although a high antiquity is claimed for the origin of astro- 
nomical science, yet it appears certain that its earliest de- 
velopment during many centuries was slow, and its results 
limited. The ancients considered the planets and the firma- 
ment to revolve in perfect circles, the centre of which they 
long supposed to be occupied by the earth, and not by the sun. 

The true doctrine, that the sun is the centre, was first 
taught by Pythagoras, in ' the sixth century before Christ, or 
more than twenty-three centuries ago. But as it was com- 
municated privately to his disciples, and in the absence of the 
modern facilities of printing and intercourse, it is probable 
that it was not very widely adopted. Seven centuries after- 
ward, the doctrines of Ptolemy again made the earth the 
centre of the system, which error prevailed during fourteen 
centuries, till the true theory was revived by Copernicus, 
three centuries ago, in 1543, by the publication of his work, 
" De Revolutionibus Orbium Celestium," wherein he considers 
the sun to be the centre of the solar system, as taught by 
Pythagoras. 



PLATE NO. P. 91 

Copernicus appears to have believed in the perfectly circu- 
lar form of the celestial orbits, as taught originally by Pytha- 
goras, and his successors of the peripatetic school. 

The half century succeeding Copernicus, gave birth to the 
three greatest of modern astronomers. Tycho Brahe, in 1546, 
Galileo, in 1564, and Kepler, in 1571. 

Tycho Brahe, born in 1546, at Knudsthorp, and educated 
at the University of Copenhagen, having witnessed when only 
thirteen years of age the occurrence of a great eclipse of the 
sun, and having seen the sun darkened at the very moment, 
and to the very extent which had been predicted and deline- 
ated, directed his attention to astronomy, and with great zeal, 
and under all discouragements, continued to devote himself to 
his favorite science. 

When Tycho was about thirty years of age, his sovereign, 
Frederic II, of Denmark, a prince distinguished for liberal and 
scientific tastes, gave him a grant of a convenient site, the 
island of Huen, and erected an observatory and accommoda- 
tions on a liberal scale, at an expense of one hundred thousand 
dollars, and Tycho is said to have expended on them a similar 
sum from his private fortune. The observatory and instru- 
ments erected under Tycho's own superintendence, were on a 
scale which so far surpassed in size, in ingenuity of construc- 
tion, and efficiency, any which had previously been used, 
that his observations approached within a few seconds of 
absolute accuracy, instead of a few minutes, as had hitherto 
been practicable. 

Here for more than twenty years, Tycho devoted himself 
to observations of the celestial motions with great zeal and 
success. 

It may be remarked, that the three great epochs of astrono- 
my, as a science of pure observation are 

First, that of Hipparchus, who measured the earth from the 



92 EXPLANATION OF THE PLATES. 

heavens, and who has been called the founder of Astronomy, 
and of correct geography, 160 years before Christ, whose 
observations approached to correctness within one half degree. 

Second, that of Tycho Brahe, A. D. 1580 to 1600, correct 
within two minutes, or A of a degree. 

The first use of the telescope, Galileo, January, 1610. 

Third epoch of observation is that of Bradley, author of 
the discoveries of Nutation, Aberration, and of improvement 
in the tables. of refraction, 1742 to 1762. 

Bradley's Observations, observing with the long telescope, 
of 212 feet focal length, introduced by Huygens, are correct 
within two seconds, or yAu of a degree. 

Sir David Brewster observes : " as a practical astronomer, 
Tycho has not been surpassed by any observer of ancient or 
modern times. The splendor and number of his instruments, 
the ingenuity which he exhibited in inventing new ones, and 
in improving and adding to those which were formerly known, 
and his skill and assiduity as an observer, have given a char- 
acter to his labors which will be appreciated to the latest 
posterity." 

The death of Frederic in 1588, and the accession of a new 
dynasty, after various ungracious incidents, at length sent 
Tycho, in 1597, an involuntary exile from his ungrateful 
country, and by the invitation of Rudolph, emperor of Bohe- 
mia, he visited Prague, where he was kindly received, his 
immediate wants provided for, and the castle of Benach, a few 
miles distant, was allotted for his home, with a pension of 
three thousand crowns. 

On this occasion, an exile from his country, and the spot 
where he had devoted so much labor, he comforted himself 
with the idea that every soil was the country of a great man, 
and that wherever he went the blue sky would still be over 
his head ; and dying in exile, during the temporary delirium 



PLATE NO. 9. 93 

which accompanied his last illness, frequently exclaimed, "Ne 
frustra vixisse videor." It is certain I have not lived in vain. 

In January, 1600, Kepler paid a visit to Tycho, who 
received him in the kindest manner, and gave him an intro- 
duction to the emperor, and after remaining three or four 
months, returned to Gratz. But he returned in September, 
1601, Tycho having obtained for him the appointment of im- 
perial mathematician, on the condition of Kepler assisting him 
in his observations, and on Tycho's death, which happened in 
October of the same year, Kepler succeeded him as principal 
mathematician, and to the inheritance of his valued observa- 
tions. It was thus that fortuitous or providential events, the 
persecutions of the age which exiled Tycho, and which sub- 
jected Kepler to continued poverty, and the proffered hospi- 
tality o£ Rudolph, threw these two great men into association. 

Although it is certain that for a number of years previous, 
Kepler had had under consideration various hypotheses con- 
cerning the number, distance, and periodic times of the planets, 
it is equally certain that this connection was peculiarly valua- 
ble to Kepler, as the observations of Tycho were the only 
ones in the world which could enable him to carry on his own 
theoretical inquiries, and it may be that to this circumstance 
is mainly owing his grand discoveries. 

The system which Tycho had been taught, and which he 
.advocated, was identical with the one which Copernicus had 
rejected, and consisted in supposing the sun to revolve about 
the earth, carrying with it all the other planets revolving about 
him, and indeed Tycho not only denied the revolution of the 
earth about the sun, but likewise its diurnal revolution on its 
axis. 

Galileo, also, when a very young man, at the time when the 
Copcrnican system began to be discussed, and when Wurtei- 
sen, a disciple of Copernicus, delivered on this subject two or 



94 EXPLANATION OF THE PLATES. 

three lectures, was unwilling to be present, being firmly per- 
suaded that this opinion was a piece of solemn folly. And it 
is stated by Gerhard Voss, that subsequently, a public lecture 
of Maestlin, the instructor of Kepler, was the means of making 
Galileo acquainted with the true system of the universe. 

Lord Bacon, also, who was nearly a cotemporary with 
Galileo, positively condemned the idea of the earth's motion as 
absurd and false, and, it is believed, never altered this opinion. 

But Kepler, who was a few years younger than these, at 
the college of Tubingen, under the tuition of Maestlin, had 
become acquainted with the true system of the universe, and 
delighted in the simplicity of the Copernican system. 

Such is a brief outline of the theory of the universe before 
the time of Kepler. 

The theory adopted by Kepler in the early part of his 
discussion of Tycho's observations, appears to have been that 
the orbit of each planet, including the earth's was circular, and 
that it was described with an uniform angular velocity, like 
the hands of a clock, and that not only its angular but also its 
linear motion in its orbit was uniform. 

In proceeding to discuss the orbit of Mars. The observa- 
tions of Tycho had determined the motions of this planet to a 
great degree of accuracy, and Kepler soon found that the 
inequalities of motion differed widely from any thing which 
could be represented by the system of epicycles, as previously 
suggested by Copernicus. He accordingly, with incredible 
diligence, directed his observations to investigate the real form 
of the orbit of Mars. 

This was difficult, because the observations were made 
from the earth, itself in motion. Hence it was necessary to 
distinguish that apparent part of the inequality of the motion 
of Mars, which is caused by the inequality of the earth's own 
motion in its orbit. 



PLATE NO. 9. 95 

After four years of laborious calculation, the non-accord- 
ance of his adopted theory of a circular orbit with his own 
observations, and those so carefully recorded by Tycho, had 
determined that the motions of Mars were not circular, that 
they were not alike at all portions of its orbit, that it moved at 
some times more rapidly than at other times, and he was, 
therefore, obliged to seek some other explanation. 

Considering, therefore, the earth's orbit as circular, " Kepler, 
after many fruitless speculations, in endeavoring to explain 
these planetary inequalities, was led to suppose the orbit of the 
planet to be oval, and from his knowledge of the conic sections, 
he afterwards determined it to be an ellipse, with the sun 
placed in one of its foci. He then ascertained the dimensions 
of the orbit, and by a comparison of the times employed by 
the planet to complete a whole revolution, or any part of one, 
he discovered that the time in which Mars describes any 
arches of his elliptic orbit were always to one another as the 
areas contained by lines drawn from the focus, or the centre 
of the sun, to the extremities of the respective arches, or in 
other words, that the radius vector, or the line joining the Sun 
and Mars described equal areas in equal times. By examining 
the inequalities of the other planets, he found that they all 
moved in elliptic orbits, and that the radius vector of each 
described areas proportional to the times. These two great 
results are known by the name of the first and second laws of 
Kepler. The third law, or that which relates to the connec- 
tion between the periodic times and the distances of the 
planets, was not discovered till a later period of his life." 

[Sir David Brewster. 

What is "the radius vector?" Let us be certain. The 
radius is a straight line from the centre to the circumference 
of a circle ; vector (from veho) means carried or moved ; the 
radius vector is the radius moving, the radius revolving, the 



96 EXPLANATION OF THE PLATES. 

revolving radius. Now the radius of the first circle, one inch 
in length, sweeps in one revolution the complete area of this 
circle. The rule is universal. The radius of any circle 
sweeps in one revolution its complete area. The area of the 
seventh circle is sixty-four times that of the first. It is obvious, 
therefore, that a planet revolving in an orbit represented by 
the first circle, to describe " equal areas in equal times," must 
revolve exactly sixty-four times, while another planet, revolv- 
ing in an orbit represented by the seventh circle, revolves 
once. 

We have already demonstrated that the intensity of light 
and of heat at distances. represented by the radii of the respec- 
tive circles, are as the length multiplied into the breadth of 
those radii. We have assumed, and it is an important truth 
we maintain without fear of contradiction, that all radiant 
phenomena illustrate each other. Centripetal phenomena in- 
versely represent cetrifugal. 

We have shown (page 88,) that the centrifugal force or 
motion is equal to, and is compensated, balanced, indicated, 
measured by an equal centripetal force or gravity, at any 
distance respectively. 

If this be true, the spaces swept over by each one of any 
number of bodies, revolving by the law of gravity, at various 
distances, whether these bodies revolve in circles or in ellipses, 
will represent the times respectively occupied by each. The 
spaces in truth represent times. The spaces are simply 
indexes to the times. The hand moving over the dial of a 
clock is a radius, it is a radius vector, and it sweeps equal 
areas in equal times. 

Let the centre of the circle in Plate No. 9, represent the 
Sun. Let the first circle represent the orbit of Mercury, the 
second that of Venus, the third that of the Earth, the fourth 
that of Mars, the fifth Jupiter's, the sixth Saturn's, the seventh 



PLATE NO. 



97 



H3rschell's. It is obvious that to sweep equal areas, the first 
must revolve twice while the second revolves once ; four 
times while the third revolves ; eight times while the fourth ; 
sixteen times while the fifth ; thirty-two times while the sixth, 
and sixty-four times while the seventh revolves once. 

Plate No. 9 may also be made a means of explaining the 
principles of the laws of Gravity, as exemplified in the move- 
ment of that most useful and interesting invention, the pen- 
dulum. 

The motion of the pendulum is in a plane. It is like the 
planetary revolutions, the result of two forces. One of these 
forces is gravity, (centripetal,) which inclines the pendulum 
directly to the earth's centre, the other is a lateral (centrifugal) 
force, produced mechanically by a spring or weight, &c. 
The planetary motions are celestial mechanics. The motion 
of the pendulum is terrestrial mechanics. But the great truth 
of the combined action of two forces explains both. * 

The planets, at different distances sweep equal areas in 
equal times. Pendulums of different lengths sweep equal 
areas in equal times. Plate No. 9 equally well exemplifies 
both. If we shorten the length of any pendulum in the least, 
or to any degree, we shall increase its lateral motion, or its 
speed to a compensating extent. If we shorten a pendulum 
to half the length, either its motion laterally, or its speed, (or 
the resultant of both,) will be doubled ; if to one quarter the 
length, quadrupled. If we lengthen it, the reverse. Thus, if 
we have a pendulum of eight inches in length, that is, whose 
vibrations are represented in the Plate by the section or 
colored portion of the largest circle of eight inches radius, and 
if we shorten the radius, (the length) one half (to four inches,) 
or three quarters (to two inches radius,) we shall correspond- 
ingly increase the breadth. See Plate No. 9. 

The author of the pendulum was Galileo, and perhaps a 



98 EXPLANATION OF THE PLATES. 

short digression from the continuity so important to our subject, 
may be allowed, as an historical interlude. We will quote 
from Prof. Powell : 

"A few years earlier than Kepler, (Feb. 15, 1564,) was 
born his illustrious friend and cotemporary in public fame, 
Galileo de Galilei, the son of Vicenzo de Galilei, of Pisa, in 
Italy. After the fashion of that age and country, he appears 
to have been more commonly designated by the name of 
Galileo, by which he has been since known. 

" Vicenzo was a man of considerable learning and talent, 
who early appreciated and properly cultivated the expanding 
genius of his son. At the university of Pisa he was very soon 
distinguished, not merely for general proficiency in his studies, 
but for the singular boldness with which he maintained original 
opinions, often at variance with the received scholastic 
dogmas. 

" One grand tenet of the schools was that heavy bodies fall 
to the earth more rapidly than lighter, in proportion to their 
weight. Galileo, in the presence of the university, ascended 
the leaning tower of Pisa, and dropped from its summit bodies 
of different weights ; with an inconsiderable difference, due to 
the resistance of the air, they reached the ground nearly at 
the same moment. The learned quoted Aristotle in preference 
to their senses, and contended that a mass of ten pounds must 
and does fall in one tenth of the time occupied by the falling 
of one pound, and as Galileo had made them fall in the same 
time, the only result was an inveterate hostility against him. 

"In the year 1592, Galileo published a treatise "Delia 
Scienza Mecanica," in which, after examining the theory of 
the mechanical powers, he lays down the general proposition, 
that all the advantage given by them is simply this : they make 
a small force equivalent to a great one, by causing it, the 
former, to move over a proportionally greater space in the 



PLATE NO. 9. 99 

• 

same time. And in following out this principle, he further 
shows, that if the effect of a force be estimated by the weight 
it can raise to a given height, in a given time, this effect can 
never be increased by any mechanical contrivance whatever. 
He proceeded also to the theory of the oblique lever and the 
inclined plane. This was the discovery of the well known 
truth, that what is gained in time is lost in power, and vice 
versa. 

" In the cathedral of Pisa, one of the chandeliers hanging 
from the lofty roof had accidentally been set swinging : this 
instantly struck the observant mind of the young philosopher, 
who noticed the fact that its vibrations were performed in 
exactly equal times, by comparing them with his pulse. This 
subject afterwards occupied much of his attention, but for the 
present he went no further than to apply the principle to the 
purposes of the medical profession, to which his views were 
directed, by contriving a pendulum with a variable length of 
string, by which its beats might be made to accord with those 
of the pulse, and thus give a measure of its rate. 

" In the theory of motion he had, as we have seen, investi- 
gated some of the laws of falling bodies. He extended this 
inquiry, by first assuming that they receive equal increments 
of velocity in equal times, and thence deduced mathematically 
that the spaces described must be as the squares of the times, 
and that the space fallen through in one portion of time is 
exactly half that which would be described in the same time, 
with the velocity last acquired continued uniformly. He soon 
saw that a body descending on an inclined plane must be in 
like manner accelerated ; he therefore adopted this as a simple 
mode of putting his theory of falling bodies to the test of 
experiment. 

" From his knowledge of the inclined plane, it easily fol- 
lowed that the times of falling down, all the chords of a circle, 



6o 



EXPLANATION OF THE PLATES. 



terminating at its lowest point, must be the same. He how- 
ever fell into an error in maintaining the same of the circular 
arcs, and in applying this to the vibrations of the pendulum. 

" Knowing the law of falling bodies, to give spaces propor- 
tional to the squares of the times, he deduced the motion of 
projectiles, and showed their path to be a parabola. This 
reasoning involved a principle which Galileo does not expressly 
refer to by name, but which is, in fact, one of the cases of the 
general principle of the composition of forces. 

" The mechanical and dynamical researches of this great 
philosopher are unquestionably those in which we recognise 
the first real union of experimental and mathematical reason- 
ing in investigating the laws of force and motion. 

" These researches, although perhaps less brilliant than his 
astronomical discoveries, on which his popular fame may be 
more universally founded, are of inestimable value." 

The following able summary of his character and labors is 
given by Prof. Playfair : 

" One forms, however, a very imperfect idea of this philoso- 
pher from considering the discoveries and inventions, numerous 
and splendid as they are, of which he was the undisputed 
author. It is by following his reasonings, and pursuing the 
train of his thoughts, in his own elegant, though somewhat 
diffuse exposition of them, that we become acquainted with 
the fertility of his genius, with the sagacity, penetration, and 
comprehensiveness of his mind. The service which he ren- 
dered to real knowledge is to be estimated, not only from the 
truths which he discovered, but from the errors which he de- 
tected ; not merely from the sound principles which he estab- 
lished, but from the pernicious idols which he overthrew. 

" His acuteness was strongly displayed in the address with 
which he exposed the errors of his adversaries, and refuted 
their opinions by comparing one part of them with another, 



PLATE NO. 9. 101 

and proving their extreme inconsistency. Of all men who 
have lived in an age which was yet only emerging from 
ignorance and barbarism, Galileo has most entirely the tone of 
true philosophy, and is most free from any contamination of 
the times in taste, sentiment, and opinion. 

" By the writings of .Copernicus, Kepler, and Galileo, the 
solar system, and the subordinate series of truths referring to 
the theory of motion, were so completely established, and 
universally demonstrated, that nothing was wanting but time 
to allow men's opinions to come gradually round to the truth. 

" The lengthened career which Providence assigned to 
Galileo was filled up throughout its rugged outline with events 
of dramatic interest. But though it was emblazoned with 
achievements of transcendent magnitude, yet his noblest dis- 
coveries were the derision of his cotemporaries, and were 
even denounced as crimes which merited the vengeance of 
heaven. 

" His scientific character, and his method of investigating 
truth, demand our warmest admiration. The number and 
ingenuity of his inventions, the brilliant discoveries which he 
made in the heavens, and the depth and beauty of his re- 
searches respecting the laws of motion, have gained him the 
admiration of every succeeding age." 

In exemplification of the Third Law of Kepler, let us once 
more refer to Plate No. 9. 

This third law, as he himself informs us, first entered his 
mind on the 8th of March, 1618, but having made an erroneous 
calculation, he was obliged to reject it. He resumed the 
subject on the 15th of May, and having discovered his former 
error, he recognised with transport the absolute truth of a 
principle, which, for seventeen years had been the object of 
his incessant labors. The delight which this grand discovery 
gave him had no bounds. " Nothing holds me," says he : " I 



102 EXPLANATION OF THE PLATES. 

will indulge in my sacred fury. # * * "If you 
forgive me, I rejoice ; if you are angry, I can bear it. The 
die is cast ; the book is written, to be read either now, or by 
posterity, I care not which. It may well wait a century 
for a reader, as God has waited six thousand years for an 
observer." 

What is the third Law 1 Let us recapitulate. The first 
Law of Kepler is, that the orbits of the Planets are not 
circular. That they are ellipses, of which the sun occupies 
one of the foci. 

The Second Law of Kepler is, that the radius vector of a 
planet, and also that the radii vectores of a number of planets 
at various distances, describe equal areas in equal times. 
This law we have exemplified on pages 95 and 96, viz : That 
the orbits of the planets are planes. That they have length 
and breadth only. That they can belong only to the second 
class of measurements. 

The Third Law of Kepler is, that the squares of the 
periodic times are to one another as the cubes of the distances 
from the sun. The cubes of the distances '! This indicates 
Length, Breadth, and Thickness. This is obviously the third 
class of measurements. How is this 1 

The orbits of the planets are planes. That is certain. But 
Gravity, acting from the sun is not a mere plane. It is acting 
on all sides spherically. It is diffusing itself in spheres. Its 
intensity, like that of light and heat diminishes as the length, 
breadth, and thickness of its radii of diffusion, as the cubes of 
the distances. True ! true ! that 's it. It belongs to the third 
class of measurements. 

It might be expressed thus : The plane circles or ellipses in 
which the planets are revolving, are to each other as the 
corresponding spheres in which gravity exerts or diffuses 
itself. 



PLATE NO. 9. 103 

Or, the squares of the numbers which express the times of 
revolution, are to one another as the cubes of those numbers 
which express the mean distances of the planets from the sun. 

Or, the length and breadth (the squares) of the plane orbits 
(circular or elliptical) in which the planets are revolving, are 
to each other as the length, breadth, and thickness of the 
spheres in which gravity exerts its intensity. 

Let us observe Plate No. 9. Let the centre of the circles 
represent the sun, and the various circles represent orbits of 
the planets. Let each of the circles represent a sphere in 
which gravity is exerting itself, and around which the orbit of 
the planet is described. Circles are to each other as the 
length multiplied by the breadth of their radii ; that is, as the 
squares of their radii. Spheres are to each other as the 
length multiplied into the breadth, and again this product 
multiplied by the thickness ; that is, as the cubes of their radii, 
the cubes of the distances. In this view, Plate No. 9 is a mere 
section of a nest of hollow spheres fitting into each other. 

Sir John Herschell thus treats of this subject, Chap. VIII, 
page 262 : 

" In casting our eyes down the list of the planetary distances, 
and comparing them with the periodic times, we cannot but 
be struck with a certain correspondence. The greater the 
distance, or the larger the orbit, evidently the longer the 
period. The order of the planets, beginning from the sun, is 
the same, whether we arrange them according to their dis- 
tances, or to the time they occupy in completing their revolu- 
tions, and is as follows : Mercury, Venus, Earth, Mars, the 
four zodiacal planets; Jupiter, Saturn, and Uranus. Thus, the 
period of Mercury is about 88 days, and that of the Earth 
365, being in proportion as 1 to 4 .15, while their distances 
are in the less proportion of 1 to 2 .56, and a similar remark 
holds good in every instance. Still the ratio of increase of the 



104 EXPLANATION OF THE PLATES. 

times is not that of the squares of the distances ; it is not so 
rapid. 

" The square of 2 .56 is 6 .5536, which is considerably 
greater than 4 .15. An intermediate rate of increase between 
the simple proportion of the distances, and that of their 
squares, is thus clearly pointed out by the sequence of the 
numbers, but it required no ordinary penetration in the illus- 
trious Kepler, backed by uncommon perseverance and indus- 
try, at a period when the data themselves were involved in 
obscurity, and when the processes of trigonometry and of 
numerical calculation were encumbered with difficulties, of 
which the more recent invention of logarithmic tables has 
happily left us no conception, to perceive and demonstrate the 
real law of their connection. 

This connection is expressed in the following proposition : 
" The squares of the periodic times of any two planets are to 
each other in the same proportion as the cubes of their mean 
distances from the sun." Take for example, the Earth and 
Mars, whose periods are in the proportion of 365 .2564 to 
686 .9796, and whose distances from the sun are in the pro- 
portion of 100 .000 to 152 .369, and it will be found by any one 
who will take the trouble to go through with the calculation, 
that (365 .2564) 2 : (686 .9796) : : (100 .000) 3 to (152 .369) 3 " 

[The expression of this law of Kepler requires a slight modi- 
fication, when we come to the extreme nicety of numerical 
calculation, for the great planets, due to the influence of their 
masses. This correction is imperceptible for the Earth and 
Mars.] 

Sir John Herschell proceeds : 

" Of all the laws to which induction from pure observation 
has ever conducted man, this third law (as it is called) of 
Kepler may justly be regarded as the most remarkable, and 
the most pregnant with important consequences. When we 



PLATE NO. 10. 105 

contemplate the constituents of the planetary system from the 
point of view which this relation affords us, it is no longer 
mere analogy which strikes us, no longer a general resem- 
blance among them, as individuals independent of each other, 
and circulating about the sun, each according to its own 
peculiar nature, and connected with it by its own peculiar tie. 
The resemblance is now perceived to be a true family like- 
ness ; they are bound up in one chain, interwoven in one web 
of mutual relation and harmonious agreement, subjected to 
one pervading influence, which extends from the centre to the 
farthest limits of that great system, of which all of them, the 
earth included, must henceforth be regarded as members." 



PLATE No. 10. 

Plate No. 10 docs not differ in principle from No. 9; the 
circles are the same, and the colored portions arc similar in 
area, and only differently arranged. It is designed in aid of 
No. 9, further to exemplify the same truths, the laws of 
Kepler, and the generalization of these laws by Cassini, and 
by Sir Isaac Newton. 

The third Law of Kepler was discovered on the 15th of 
May, 1618. Half a century later, these laws of Kepler 
received a most beautiful confirmation in their application to 
the motions of the satellites of Jupiter and Saturn, from the 
careful observations of the elder Cassini, who was invited 
from Italy into France by Louis XIV, in 1669, and established 
in the observatory at Paris, where he continued a long series 
of accurate and valuable labors, particularly directing his 
attention to perfecting the theory of the interesting system of 
Jupiter, which so beautifully represents, on a small scale, the 
greater planetary system. He also discovered, ultimately, 



10G EXPLANATION OF THE PLATES. 

four satellites of Saturn, in addition to the one observed by 
Huyghens. 

Another beautiful verification was afforded, eleven years 
later, by the observations of Newton, on the brilliant comet of 
1680, which determined that its orbit was a parabola, and 
that its rate of motion also accorded with this law of the 
equable description of areas in equal times. Comets range 
into our system from all quarters of the heavens, and from 
distances inconceivably great in the depths of space. Hence 
gravitation extends in all directions, and to unknown and 
inconceivably great distances from the centre of our world. 

But it was two years later, in 1682, that Newton applied 
the same laws to the motion of the Moon round the Earth, 
and its connection with that particular physical property 
which we find belonging to particles of matter on the earth's 
surface, called gravity, or weight, which application consti- 
tutes his most valuable and remarkable achievement. 

The orbit and distance of the moon had been so well ascer- 
tained by the long continued labors of astronomers, that with 
respect to them there could be little doubt. Not so the magni- 
tude of the earth. Picard's more accurate determination of 
the earth's measurement, founded upon arcs of the meridian, 
was the subject of discussion at a meeting' of the Royal 
Society, in June, 1682. 

The raging of the plague in 1665-6, induced Newton to 
quit Cambridge, and retire to Woolsthorpe in Lincolnshire, his 
native parish. It was here that he began to reflect upon the 
nature of the force by which bodies at the earth's surface are 
drawn towards its centre, and to conjecture that the same 
force might possibly extend to the moon, and there be of suffi- 
cient intensity to counteract the centrifugal force of that 
satellite, and thereby retain it in its orbit about the earth. To 
compare this hypothesis with observation, it was necessaiy to 



PLATE NO. 10. 107 

determine the law according to which the intensity of such a 
force would vary with the distance from the earth's centre. 
In other words, simply to apply the law of Kepler, and thus 
generalise its application further than had yet been done. 
But Newton, at that time, in 1G6G, failed to establish the truth 
of this theory of gravitation, in consequence of his employing 
an erroneous measure of the earth's radius — a degree of lati- 
tude being then estimated at GO miles instead of G9^, its more 
correct length. Years passed on. It was about 1G79, when 
Picard, one of the most ingenious and faithful of observers, 
(originally a gardener) instructed by the astronomers Le 
Valois and Gassendi, and who furnished the plan of the 
observatory of Paris, made the first application of the tele- 
scope to the measurement of terrestrial angles. The superiority 
of his instruments and methods, and the care with which his 
operations were conducted, inspired deserved confidence in his 
results. Picard, 'while his friend and cotemporary Cassini 
was applying his telescope to the determination of the orbits 
and motions of the satellites of those distant planets, Jupiter 
and Saturn, was directing his attention to the earth's measure- 
ment, and at length made a successful determination of an arc 
* of the meridian. 

It was three years after this, that, as we have mentioned, 
the subject was discussed at a meeting of the Royal Society, 
in June, 1G82, and Newton having taken a note of the result, 
and thus deduced the length of the earth's radius, eagerly 
resumed his former calculation, and having now the necessary 
clement for its verification, he revived his former hypothesis, 
which had lain dormant for sixteen years. 

These laws of Kepler were discovered before the invention 
of the telescope. The progress of this discovery, although 
Galileo had been condemned to perpetual imprisonment, and 
Kepler died in poverty and starvation,was onward; the obser- 



108 EXPLANATION OF THE PLATES. 

vation of the truth of these laws could not fail to excite the 
interest of astronomers. They were found to be true, abso- 
lutely true, but at the same time subject to some very remark- 
able temporary disturbances. 

It was found that when two planets approached nearest to 
each other, that is, when they were both on the same side of 
the sun, that they exerted upon each other a reciprocal 
attraction. Their approach towards each other, as the inner 
planets (or planets nearer to the sun) revolved more rapidly, 
was always by the nearer planet overtaking, as it were, the 
outward planet in its less rapid revolution, and it was found 
that the nearer or overtaking planet was accelerated in its 
orbit, while at the same time the overtaken planet was 
retarded. This effect was observable until the two planets 
were in conjunction, that is, until both planets were on the 
same side of the sun, and at the point of their orbits opposite 
and nearest to each other, or until (if they were revolving in 
the same plane, which they are not,) a line might be drawn 
from the sun through the centre of both planets. Now the 
reverse effect occurred; the mutual attraction still existed, but 
it was changed ; the outer planet, whose motion had hitherto 
been retarded by the gravity of the nearer or overtaking' 
planet was immediately accelerated, and the inner or over- 
taking planet was no longer accelerated, but to a corresponding 
extent retarded. And it was found that the mutual accelera- 
tion and retardation exactly balanced, and were reciprocally 
compensatory to each other, and that thus both planets were 
restored to their own true orbit and motion of equal areas in 
equal times, as though no disturbance had happened. 

In two planets (by supposition) of similar weight, the ac- 
celeration and retardation would be both compensatory and 
equal, one would be accelerated as much as the other would 
be retarded, Bat if either planet, say the accelerated planet, 



PLATE NO. 10. 109 

were in any proportion heavier than the other, say ten times 
heavier, it would retard the overtaken planet ten times as 
much as it would itself be accelerated. But the instant it 
passed the point of conjunction it would commence, in the 
same ratio the compensating process ; it would now accelerate 
the lighter one ten times as much, restoring it to its independ- 
ent rate of motion, while its own retardation would be only 
one tenth, or in a ratio precisely compensatory. 

The comparative reciprocal attraction indicated by this 
disturbance in the time of each other's motion, affords a very 
perfect means of comparing the weight of the planets, weigh- 
ing them against each other, as it were on a steelyard. 

The consciousness of this truth as regards the known 
planets, was the moans of the discovery of the last outside 
planet Neptune, which, though it is a large planet, yet from 
its vast distance, might have otherwise remained forever 
undiscovered among the fixed stars. 

The planet Herschel, or Uranus, was discovered by Sir 
William Hcrschell, in 1781. Though eighty-two years, or 
earth's revolutions have since passed, this planet has not to 
this time (1853,) completed an entire revolution round the 
sun ; its entire revolution requiring a little more than eighty- 
four years. But its path has been carefully observed by Sir 
William Hcrschell, and a numerous succession of astronomers. 
The conviction that a disturbance of the orbit of this planet 
by a more distant planet was not improbable, induced ob- 
servers to be in some expectation of such an event This ex- 
pectation was at length confirmed by the partly simultaneous 
observations of a considerable number of astronomers, of 
whom Le Verrier was the more fortunate, in being the first 
to announce to the world the existence of another and more 
distant planet, Le Verrier or Neptune. 

Le Verrier, in directing his telescope to Herschell, observing 



110 EXPLANATION OF THE PLATES. 

that its rate of motion was accelerated, and differed from the 
calculated rate as laid down by previous astronomers, at once 
inferred the existence of an exterior planet. This inference 
was confirmed by each succeeding observation, and he was 
soon able to approximate by calculation the direction of the 
disturbing planet within 54' 45" of heliocentric longitude, and 
30' 54" of latitude. He wrote to M. Galle, assistant at the 
Berlin observatory, whose means of observation were better 
than his own, informing him of this discovery. Dr. Galle 
received this letter by the post of the 23d Sept. 1846, and 
setting the Berlin equatorial telescope in the direction pointed 
out by Le Verrier, made the actual optical discovery of the 
planet on the same evening. 

The discovery of this new and remote planet affords a case 
in which the facility of application of the known laws of equal 
areas in equal times is capable of being exemplified. If while 
the earth performs an entire revolution in its orbit, this planet 
is found to have swept only one hundred and sixty-fourth part of 
the circumference of its orbit, it gives the data (provided the 
orbit be assumed as circular) for determining the distance of 
Neptune from the sun or centre of gravitation, compared with 
the distances of the other planets. 

The mutual gravitation of all bodies in proportion to their 
weight being admitted, it was evident that while the planets 
were describing their orbits round the greatest and most pow- 
erful body in the system, they must mutually attract one 
another, and thence in their revolutions some irregularities, 
some deviations from the description of equal areas in equal 
times, and from the laws of elliptic motion might be expected. 

Such irregularities, however, had not been observed by the 
earlier astronomers in the motion of any of the planets, except 
the moon, where some of them were so conspicuous as to 
have been known to Hipparchus and Ptolemy. Newton, 



PLATE NO. 10. Ill 

therefore, in consideration of this subject was very naturally 
led to inquire what the different forces were which, according 
to the laws just established, could produce irregularities in the 
case of the moon's motion. Beside the force of the earth, or 
rather of the mutual attraction of the moon and earth, the 
moon must be acted on by the sun, and the same force which 
was sufficient to bend the orbit of the earth into an ellipse, 
could not but have a sensible effect on the orbit of the moon. 
Here Newton wisely observed that it is not the whole of the 
force which the sun exerts on the moon that disturbs her 
motion round the earth, but only the difference between the 
force just mentioned and that which the sun exerts on the 
earth, for it is only the difference that affects the relative 
position of the two bodies. To have exact measures of the 
disturbing forces, he supposed the entire force of the sun on 
the moon to be resolved into two, of which one always passed 
through the centre of the earth, and the other was always 
parallel to the line joining the sun and earth, consequently, to 
the direction of the force of the sun on the earth. The former 
of these forces being directed to the centre of the earth, did 
not prevent the moon from describing equal areas in equal 
times round the earth. The effect of it, however, he showed 
to be to diminish the gravity of the moon to the earth by 
about ( one 358th part, and to increase her mean distance 
in the same proportion, and her angular motion by about 
one 179th. 

From the moon thus gravitating to the centre of the earth, 
not by a force that is exactly inversely as the squares of the 
distances, but by such a force diminished by another contra- 
vening force, it was found, from a very subtle investigation, 
that the dimensions of the elliptical orbit would not be sensibly 
changed, but that the orbit itself would be rendered moveable, 
its longer axis having an angular and progressive motion, by 



112 EXPLANATION OF THE PLATES. 

which it advanced over a certain arc during each revolution 
of the moon. Its motion was a revolution round the earth, 
itself revolving round the sun. This afforded an explanation 
of the motion of the apsides, that is, the line connecting the 
greatest and least distance of the moon from the earth, which 
had been observed to go forward, during each time of the 
moon's revolution at the rate of three degrees and four min- 
utes nearly, in relation to the fixed stars. 

In short, Kepler, in 1601, previous to the invention of the 
telescope, had achieved his great discovery of the laws by 
which the motions of the planets, including the earth, revolve 
round the sun. About 1670, Cassini, by the aid of the tele- 
scope, had applied these laws to the revolutions of the satellites 
of Jupiter and Saturn round those remoter planets. In 1680, 
Newton had himself verified the same laws in his observations 
on the elliptical orbit of the great comet of that year. In 
1682 he had only to apply the same law to the earth's satellite, 
the moon, and also to falling bodies at the earth's surface, and 
thus further to extend and to generalise what is called the 
theory of universal gravitation. 



PLATE No. 11. 

In Figure 1 we have the square inch, and the square drawn 
on its diagonal, doubling (as by the problem of Pythagoras,) 
the area of this square. It is, thus far, similar to Plate No. 8, 
the squares of which Plate are successively in the same rela- 
tion, doubling each other's areas in the ratio of 1, 2, 4, 8, 16, 
32, 64, &c, which is what is technically called a geometrical 
series. 

In Plate No. 8, the length and breadth (that is the sides,) of 
the triangles are in every case equal to each other. It is a 
simple repetition. It is now our purpose to exemplify that, 



TLATE NO. 11. 113 

however the length and breadth (that is the sides,) of the 
triangles be varied, the principle is invariable ; the diagonal 
(the hypothenuse) will in every case be a result of the length 
and breadth. It is a function of the length and breadth. 

In Plate No. 1 1 we show that we can present a series of 
squares which shall have to each other the relation in area of 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on, in an infinite series, 
which is technically called an arithmetical series. An arith- 
metical series is where the increase is always equal. 

In Figure 2 we first repeat Figure 1. If we now, on the 
right, add another square inch at right-angles with the side of 
the second square, and draw the hypothenuse, the square 
which may be drawn on this must be equal to the other two 
squares. We know these to be of one and two inches area, 
and it is evident that this resulting square is of three inches 
area. For the dissection of this square, and the re-arrange- 
ment of the three square inches on the extreme right, the 
author is indebted to an ingenious friend. 

Above, on the side of the third square, we again add, at 
right-angles with it, another square inch, and draw the hypo- 
thenuse. It is evident that the square on the hypothenuse is 
now four inches. 

Proceeding to Figure 3, we add to square No. 4, at right 
angles, another inch, and the square on the hypothenuse is 
five inches area. This square may be analysed into the one 
square inch in the middle, and the four similar triangles which 
surround it. Each of these triangles being the half of paral- 
lelograms of two inches length by one inch breadth, arc (by 
Plate No. 2,) one square inch in area. Four triangles of one 
inch and one square inch in the middle = 5. 

Figure 4 presents another example of the same invariable 
problem. To a square of three inches sides = 9 inches area, 
we add, at right-angles, a square inch, and draw the hypothe- 

o 



114 EXPLANATION OF THE PLATES. 

nuse. Our triangle is now three inches length, by one inch 
breadth. Its area is 1-J inches. The square drawn on the 
diagonal or hypothenuse, is 10 inches area. This may be 
analysed into the square of two inches sides, four inches 
area in the middle, and the four triangles or half parallelo- 
grams surrounding it. Each of these is three inches in length 
by one inch in breadth, and consequently contains l\ square 
inches area. 4X1^=6 + 4=10. 

Figure 5 presents another example. To a square of four 
inches sides, 4x4=16, we add, at right-angles, a square inch, 
and draw the diagonal. Our triangle is now four inches 
length by one in breadth. The square drawn on the hypo- 
thenuse is 17 inches area. This may be analysed into the 
square of three inches sides = 9 inches area in the middle, 
and the four surrounding triangles or half parallelograms. 
Each of these is four inches in length by one inch in breadth, 
and consequently contains 2 square inches in area. 4X2 
= 8 + 9 = 17. 

Figure 6 comprises Figures 1, 3, 4, and 5, combined in 
one. In Figure 6 each of these may be contemplated sepa- 
rately. We may also demonstrate the area of each of these 
squares from the other, either synthetically, starting from the 
smaller squares, or analytically, starting from the larger. In 
either case, the triangles added or cut off are the precise 
counterparts of the same parallelogram. 

If we start from the largest square of four inches sides, 16 
inches area, 16 inches area, less four triangles of 1^- inches 
area, = 10 inches area. Or, if we start from the middle 
square of four inches area, added four triangles of 1^ inches 
= 10 inches area. 16— (ljx4) = 10. 4 + (lJx4) = 10. If 
we start from the square inch in Figure 6, which corresponds 
to the middle square in Figure 3, it is 1 + (1X4)=5. 9 — 
(1X4)=5. 



PLATE NO. 13. 115 

PLATE No. 12. 

Plate No. 12 presents, combined in one connected figure, a 
series of examples of the same problem illustrated in Plate 
No. 11. 

On the left, starting from the square inch, we first draw the 
square on its diagonal = 2 inches area. To this, at right- 
angles, we add another square inch, and draw on the hypothe- 
nuse the third square of three inches area, and again, in the 
same manner, 3 + l=the square of four inches area; the 
fourth, 4+1 = 5; the fifth +1 = 6; the sixth +1 = 7 inches in 
area. 

To the seventh square we add, exactly as before, at the 
further side, the one square inch, and draw on the hypothe- 
nuse the square No. 8. 

We now alter the plan, but not the principle of our figure. 
On the nearer side of square No. 8 we add the one square 
inch, and draw the diagonal. On this we build square No. 9, 
of nine inches area. On the nearer side of square No. 9, 
again adding one square inch, we draw the diagonal, and 
form square No. 10 = 10 inches area. In like manner we 
form successively squares 11, 12, 13, 14, 15, 16, each of them 
increasing in area one inch upon the preceding one, and all 
of them being examples of the invariable problem of Pytha- 
goras. 

It is also obvious that we might continue to add, in a simi- 
lar manner, the one square inch without limit, thus forming an 
infinite scries of squares increasing in the same ratio. 



PLATE No. 13. 

Plate No. 13, like Plate 12, presents another scries of 
examples of the same problem of Pythagoras, forming one of 
a system of spirals which is capable of unlimited variety, 



116 EXPLANATION OF THE PLATES. 

and when fully understood will be found to have considerable 
novelty, utility, and beauty. 

Drawn one inch to the right from the centre, we have the 
square inch. On the diagonal of this square we build the 
square of two inches area, the length of the side of which is 
expressed by the square root of two, or, in short hand, >/2. 
On the near side of this square, we again add the square inch, 
and, drawing the hypothenuse, build the square No. 3, of three 
inches area, the side of which is the square root of three, 
>/2. Proceeding from square No. 3, in the same manner we 
form successively squares No. 4, 5, 6, 7, 8, and so on, up to 
16, which is all we have drawn. But it is obvious that we 
might go on in a similar manner without any limit, that the 
progression might be extended to hundreds, thousands, or 
millions, that it is capable of an infinite series. 

This diagram, with the six which follow, being valued by 
their author as important demonstrations of geometrical truths 
of which Pythagoras was the first discoverer, are with grati- 
tude inscribed to the memory of that great philosopher. 

Pythagoras flourished about 550 years before Christ. He 
was a pupil of Thales, the earliest of Greek geometers. To 
him the world is indebted for the discovery of that remark- 
able problem which constitutes the forty-seventh proposition 
of the first book of Euclid, viz : that the square described 
upon the hypothenuse is equal to the sum of the squares 
described upon the other two sides ; a demonstration of the 
highest importance, from the very numerous applications it 
finds in every branch of mathematical science. Prof. Powell 
rightly designates this the fundamental theorem of geometry, 
and justly remarks that the story related of its author's sacri- 
ficing a hecatomb for joy, on its discovery, would hardly have 
been invented if he had not at least enjoyed the reputation of 
originality in the discovery, 



PLATES NO. 14 & 15. 117 

The merits of Pythagoras were fully acknowledged by 
antiquity. When the Romans, in the 411th year of the city 
were commanded by the oracle of Delphos to erect statues 
to the bravest and wisest of the Greeks, the distinguished 
honor was conferred on Alcibiades and Pythagoras. 

To Pythagoras Philosophy is indebted for the name it bears. 
Previous to his time it had been the custom among the Greeks 
to call men of observant and reflective minds aoqog, wise; he 
took the more modest title of (pdog-aoyia, " a lover of wisdom." 

" He had studied under Pherecydas, a disciple of Pittachus ; 
but to neither of these does it appear that he was indebted for 
any new knowledge of mathematical or physical subjects. 
On his return from his travels in Egypt and the East, in the 
time of the last Tarquin, finding his native country, Samos, 
under the dominion of the tyrant Polycrates, he went as a 
voluntary exile, to seek a tranquil retreat in Italy. At Crotona, 
(as we learn from Ovid,) he studied and taught those sublime r 
views of the material universe, into which an insight is to be 
acquired by the rejection of artificial systems, and a free 
communing with nature in her own domains. Pythagoras 
seems to have clearly understood what we now distinctively 
call the solar system of the world, and to have recognised 
the diurnal rotation as well as the annual revolution of the 
earth, the central position of the sun, and the revolutions of 
the planets, to which he added a just idea of the nature of 
comets. All this, however, was communicated only to his 
disciples in private. It has also been supposed that he, or his 
immediate followers taught the probable existence of other 
systems, of which the fixed stars were the suns. 

PLATES No. 14 & 15. 
Plate No. 14, and Plate No. 15, are in truth the same Plate 
as No. 13, The shading and color used may, perhaps, have 



118 EXPLANATION OF THE PLATES. 

great utility in making a lucid impression on the eye, and on 
the minds of young and unitiated students, but it is really not 
necessary to the diagram, intellectually considered, that is if 
persons would exert their intellect ; though it may not impro- 
bably be an important aid in making the subject attractive 
and comprehensible to many for whom this treatise is de- 
signed. 

We have repeatedly stated as an axiom, that length multi- 
plied by breadth is the simple, universal, invariable law of all 
superficial figures, and we have also, in Plates 8, 9, & 10, 
made some progress in demonstrating that superficial areas 
represent, and constitute the measure of all rotary motion, 
whether in terrestrial or celestial mechanics. We say length 
multiplied by breadth. But "multiplication is but compen- 
dious addition," (see page 35.) Length and Breadth, length 
added to breadth, mean the same thing, and identify in all 
applications to mechanics, the second class of measurements. 

Logarithms, from logos, a ratio, and arithmos, a number, are 
tables exhibiting arithmetically the relation of roots and of 
their powers ; the geometrical relations of length, breadth and 
thickness ; the relations of circles or of spheres to their re- 
spective radii ; the relations of squares and of cubes to their 
respective roots. They were invented in 1614, by John Na- 
pier, and are very important and convenient in saving the 
continual labor of individual calculation. 

Plates No. 8 and 13 and those which follow, are good ex- 
amples of logarithmic series, forming spirals capable of infi- 
nite variety. 



PLATE No. 16. 
Plate No. 16 is also in outline precisely the same as No. 13. 
No. 14 and 15 are so colored as to designate the series of 



PLATE NO. 16. 119 

squares. In No. 16 the squares are neglected. It is the 
especial purpose of this diagram to show the triangles on 
which the squares are drawn. The first triangle extends in 
length from the centre one inch to the right. Its breadth, 
based on the first square inch, is also one inch. It is the half 
of a parallelogram of one inch length and one inch breadth. 
Its area is the half of one square inch. Its length is the side 
of a square inch. Its breadth is also the side of a square inch. 
The side of a square inch is the same as the square root of a 

Vlx VI 

square inch. Its area is expressed in short hand 

The second triangle has obviously for its length the diago- 
nal of the first square inch, which is the side of the square of 
two inches area. Its breadth being one inch, its area may be 

V2X VI 
expressed by ^ 

The third triangle has for its length the side of the square 

of three inches area. Its breadth being one inch, the side of 

r V3XV1 
one inch square. Its area is 

The fourth triangle has for its length the side of the square of 
four inches area, or two inches length. Its breadth, like all 

, , • • 1 T i • v^Xx/1 
the others is one inch. Its area, consequently, is -. = 

one square inch. The length of this triangle being two inches, 

its breadth one inch, its area consequently is one inch, or 

twice that of triangle No. 1 . 

v/5XVl VlXv/1 
The fifth triangle is ~ so that the scries is „ 

V2Xv/l v/3XVl 74XV1 v/5Xv/l v/GX^l 

2 2 2 2 2 

V7Xv/l >/8Xv/l v/9xVl 

n &C. &C, 



2 



120 EXPLANATION OF THE PLATES. 

For the length of the ninth triangle we have the side of a 
square of nine inches area, or three inches sides, that is, a 
length of three inches. As we have a length of three inches 
and a breadth of one inch, the area of this triangle is the one 
half of the parallelogram of 3X 1, or lj inches. 

We may thus continue an infinite series of triangles, the 
breadth of which would always be a constant and unvarying 
quantity — one inch, while the length is extending continually, 
and without limit, but by a rate of increase which is con- 
stantly diminishing, thus presenting a curious example of 
infinite series, in which also the angular measurement is 
diminishing in an analogous ratio. 

The breadth being constant, it is obvious that as often as 
the side of the increasing square becomes a square number, 
there will be added one inch in length to the triangle, and 
consequently one half inch in area ; but the series of square 
numbers is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and their 
difference being 3, 5, 7, 9, 11, 13, 15, 17, 19, increasing by 
two continually, the number of triangles intervening between 
each square number is constantly increasing in the same 
ratio. Thus, though each triangle is larger in area than the 
last, the ratio of increase is continually diminishing. For 
instance, the square of 100 is 100 X 100=10,000. 
Of 101 is 101X101 = 10,201. 
The square of 1,000 is l,000x 1,000=1,000,000. 
Of 1,001 is 1,001X1,001 = 1,002,001. 

Now, though there are 200 triangles intervening between 
the one of 100 inches in length, and that of 101, and 2,000 
triangles intervening between the one of 1,000 inches in 
length, and that of 1,001, and though each of these triangles 
is larger than the one that precedes it, yet it is obvious that 
the amount of area added, in any extent to which this series 
might be carried, is one half inch area to every additional 



PLATE NO. 18. 121 

inch in length added to the side of the square which corres- 
ponds to and measures the triangle in the series. 

PLATE No 17. 

Plate No. 17 is also in outline precisely similar to the four 
which precede it, but in addition, circles are drawn, the side 
of each of the squares forming the radius of each of the 
circles. Thus, the first circle has for its radius the side of the 
first square, the square inch ; the second the side of the square 
of two inches area, the third that of the square of three inches 
area, &c. The ratio of the squares to each other is known, 
and that the ratio of the circles to each other is the same as 
the squares, is also known from the previous demonstration. 

Plate No. 17 has the same uses as Plate No. 8. It is to 
show that the relation of the squares, and of the sides of the 
squares, the radii of the circles, and the rotary and radiant 
forces which these represent in all their applications to ter- 
restrial and celestial mechanics, are capable of being com- 
pared and thus measured geometrically, arithmetically, and 
algebraically. It is designed to aid the thoughtful student in 
forming a just and full conception of the laws of forces, and 
the mode in which the human intellect has subjected these to 
measurement. To connect, as it were, in one view, the 
geometry of Pythagoras with the levers of Archimedes, and 
the astronomy of Kepler, as well as with the arithmetical 
basis of those algebraic formulas in which the same grand 
truths of Nature have been clothed by Newton, La Grange, 
and La Place. 

PLATE No. 18. 
Plate No. 18 does not differ in outline from the five pre- 
ceding plates. In Plate No. 17 the sides of the squares form 
the radii of the circles respectively. In Plate No. 18 the 



122 EXPLANATION OF THE PLATES. 

sides of the squares form the diameters of the circles. The 
circles in Plate No. 18 are half the diameter, and consequently 
are in area one quarter of the circles in Plate No. 17. Plate 
No. 18 is merely an amplification, a variety of Plate No. 17, 
to which it is subordinate. 



PLATE No. 19. 

Plate No. 19 is an amplification of Plate No. 8 and Plate 
No. 17. It is designed to show that the principles illustrated, 
(the problem of Pythagoras, the levers of Archimides, the 
laws of Kepler,) are capable of unlimited application. 

The outline of the last six preceding plates forms the central 
series of Plate No. 19. In this, the added square is the inch 
square, added and repeated as often as may be desired, with- 
out limit in infinite series, exactly as in Plate No. 13, and the 
others. 

But the length and breadth of the added squares, or of 
any squares whatever, may be diversified indefinitely without 
altering the principle. Thus, in the inner or smaller series, 
the breadth is only half an inch. Half an inch in length, and 
half an inch in breadth is one quarter of a square inch in area, 
so that only one quarter of a square inch is added at each 
addition to the increasing square drawn on the hypothenuse. 
In the outer or larger series the breadth is 1^ inches; 1^ 
inches in length, and 1^ inches in breadth is a square of 2 J 
inches. So that in this series 2\ square inches in area is 
added at each addition to the increasing square. 

Wherever and whenever we have any breadth we have 
area. If we have no breadth, nothing but length represented 
by a mere line, we have no area, and no hypothenuse. The 
hypothenuse represents length and breadth. It is a function 
of length and breadth. The hypothenuse also represents the 



PLATE NO. 19. 123 

combination of (composition of) two forces, as the centri- 
petal and centrifugal force in the planetary motions. The 
composition of two forces geometrically exhibited is but 
length and breadth. The exposition of radiant phenomena is 
represented by length, breadth and thickness. It is in this 
connection, its relation to terrestrial and celestial mechanics, 
as forming the means of subjecting terrestrial and celestial 
motions to arithmetical comparison, that the study of the hy- 
pothenuse, the immortal problem of Pythagoras, the funda- 
mental problem of geometry, acquires its real importance. 

It is capable of unlimited application and of infinitesimal 
measurement, of infinitely minute as well as of infinitely 
extended applications to the phenomena of the material 
universe. 

Having thus attempted in the best manner that seems at 
present practicable, to lead the reader step by step from the 
simplest elementary considerations of measurement, to a grand 
and general conception of celestial mechanics, the author will 
here introduce an original poem, which, although its subject 
is Light, and not Gravitation, yet from the analogy which 
exists in the law of their distribution, may be of service to 
many in acquiring a just conception of the grand area in 
which the Author of creation has established the range of 
universal Light, as well as of universal Gravitation. 



THE CREATION OF LIGHT. 

And God said : " Lot there be Light, and there was Light." 

Light is certainly the most magnificent prototype which physical creation affords of the 
omnipresence of Deity, and of the unity of that self-existent universal energy, which, while it 
"spreads undivided," and "operates unspent" in the worlds of matter and of mind, is con- 
tinually revealing to intelligent creation the infinity and benignity of creation's Author. 

This sublimated ethereal substance, according to tho inspired Hebrew historian, was spoken 
into existence by tho first fiat of tho Eternal, and originated in the incipient act of creative 



124 EXPLANATION OF THE PLATES. 

energy on rude and darkling chaos. Holding, as it were, an intermediate relation to matter 
and immaterial existence, and triumphing in the inconceivahle velocity and infinite extent of 
its emanations over time and space, it may be contemplated as a sort of angelic messenger 
from the throne of the Omnipotent to the boundlessness of universal creation. 

Huygens, a Dutch astronomer of the seventeenth century, suggested that there might be 
f e stars at such an immense distance, that their light had not yet travelled down to us since the 
Creation." 

This hypothesis has not been disproved ; it is in fact sustained by the observations which 
the elder Herschel, and the more modern astronomers, by means of their gigantic telescopes, 
have been able to make in the remote ultra-planetary space. 

The idea of Huygens pictures but a solitary pencil of light, travelling from some immensely 
distant star for six thousand years, at the velocity of twelve millions of miles per minute, but 
not yet having reached us, to make its origin visible at our Earth. It is only generalizing 
this idea to believe, that the first created light of our sun, and that of every other star in the 
firmament, still extends itself in all directions into illimitable space ; maintaining, in all its 
wondrous inter-radiations, the distinct proportionate visibility of its innumerable suns, at any 
and every point of infinite space where the eye of an observer or the lens of a telescope can 
be imagined to exist. 

I. 

Instant, o'er heaven, from every sun and star, 

Flash'd the centrifugal ethereal Light ; 
The planets glittered in their paths afar, 

And darkness vanished at the dazzling sight. 

II. 

And thence far — far that darkness hastes away, 

Where our sun's starlight dawns in boundless space j 

And many a distant sun's primeval ray 
Toward us is speeding on its pauseless race. 

III. 

A hundred millionth part of Solar light 

Illumes our planets in their kindred race ; 
All, save this fraction of its radiance bright, 

Glances beyond them into boundless space. 

[The portion of the sun's radiant light which is intercepted by and received upon the earth, 
is less than one twenty-three hundred millionth of the whole. It would require more than 
twenty-three hundred and thirteen millions of such orbs as our earth to form a hollow sphere, 
the radius of which would be the earth's distance from the sun. 

[The proportion of the whole sphere of radiant sunlight received upon the disks of all of the 
planets is less than one two hundred and thirty millionth. It follows that two hundred and 
twenty-nine millions nine hundred and ninety-nine thousand nine hundred and ninety-nine 
similar portions of the whole sunlight pass beyond the limits of the solar eystem, inter-radiat- 
ing in illimitable space with the light of innumerable other suns, the fixed stars of the 
firmament. 



PLATE NO. 19 125 

[If "nothing is wasted," the purpose of the whole sunlight, (except that comparatively 
inconsiderable infiinitesimal portion, the one two hundred and thirty millionth, which is 
caught on the surface of the planets,) must be sought in the universe beyond the solar system, 
where, reciprocating with the light of other suns innumerable, it may have the more intel- 
lectual utility of revealing the existence of our sun, the unity of the universe, and the omni- 
presence of the Creator to the (supposed) inhabitants of other systems, the central suns of 
which, anchored in infinite space, appear to our limited powers of vision among the remotest 
of the telescopic stars. 

IV. 

*• 
What is our little planet's day and night ? 

The day, that dazzle's human eyes with glare. 
Checks not the interglance of starry light — 

It beams incessantly forever there. 

[The stars, except those near the sun, are at all times in very clear daylight mirrored in 
the limpid waters of deep wells; they are also visible at mid-day, reflected in the mountain 
lakes of northern latitudes, and especially when lofty and precipitous banks exclude all but 
the vertical light. The telescope, a chimney, or any opaque tube which entirely excludes the 
surrounding light, makes the stars visible from similar causes. When an observer, of a 
sunny day, shades his eye by placing his hand over it, it is to exclude the more direct rays, 
and thus to enable the less strong reflection of the distant landscape to be more visible. The 
tin tubes, blackened on the inside, which wc use in looking at pictures, are useful for the 
same reason. 

[ The use of tubes, extending by uniting them to a great length, was known to Ptolemy 
and the ancients, but Metius, of Alcmaer in Holland, was the first who added glasses, thus 
inventing telescopes in 1G09, which were successively improved by Galileo, Huygcns, Newton, 
and others, to the time of Ilerschel.] 

V. 

Wc roll into Earth's shadow ; and 'tis night 

To us — our side the orb is from the sun ; 
One half s in shadow always — one in light ; 

Evening and morning since the world begun. 

[One half of the earth's convex landscapes being always in light, and the other in shade, tho 
earth's daily rotation on its axis rolls us into and out of its own shadow, causing day and 
night. The earth, rolling from West to East on its own axis, it is always mid-day and mid- 
night at opposite meridians of longitude, and an eternal sunrise and sunset at tho intcrme. 
diato meridians.] 

VI. 

One half in day and summer light 

The eternal changeful seasons turn ; 
One half in shade and winter night 

The varying landscapes freeze and burn. 
[ Tho preceding stanza illustrates the relations of the sun's light caused by tho earth's daily 
motion on its axis. This stanza refers to those changes which result from tho earth's annual 



126 EXPLANATION OP THE PLATES. 

motion round the sun, and are caused by the inclination of the earth's axis to the plane of 
its annual path in its orbit, (the Ecliptic.) As only one half of the earth's convex landscapes 
can be in sunlight, it is obvious that day and night, and summer and winter, in all^ their 
varieties, must alternate constantly in the alternate hemispheres. As on a given parallel of 
latitude, that is on a line drawn East and West round the earth, mid-day, sunset, midnight* 
and sunrise, are perpetually present ; so of a line drawn North and South round the earth, 
one half must be in sunlight, and the other in shade. There can be but one point where the 
sunlight falls vertically from the zenith, on the earth's surface. This point is always in the 
centre of the illumined portion of the earth, varying, however, constantly with the earth's 
revolution both on its axis and in its orbit round the sun. A spiral line is indicated by 
this point on the earth's surface, which, in three hundred and sixty-five days, has extended 
from Tropic to Tropic and back again, completing the course of the Seasons in both hemi- 
spheres. About six thousand miles, or one fourth of the earth's convex, from this vertical 
point, West, it is sunrise, six thousand East, it is sunset, and also six thousand miles North 
or South from this vertical ray, the sunlight must be falling parallel with the earth's surface, 
on the icy deserts, both of the Arctic and Antarctic latitudes. 

VII. 

Myriads of sunbeams interweave their light, 
Throughout the boundless distance of the sky ; 

And gem the spangled canopy of night, 
Where'er the wanderer turns his thoughtful eye. 

VIII. 

The furthest starbeam's telescopic flight, 

Direct, unerring from its centre runs ; 
Threads the vast maze of inter-radiant light 

Athwart the day-source of a thousand suns. 

IX. 

On every side each distant sun displays, 

Across the daylight of each other sun, 
Its radiant sphere of still expanding rays, 

Widening and widening e'en since time begun. 

X. 

The feeblest starbeam in its furthest flight, 

Across the dazzling day-fires of the sky, 
Truly reveals its centre and its source 

Unscorched and changeless to the gazer's eye. 

XI. 

E'en when the student swings great Herschel's lens, 
Measuring in mighty tracts concentric space ; 



PLATE NO. 19. 127 

Each telescopic sun his vision kens 

Further and feebler has its certain place. 

XII. 

In Cancer or in Leo, as we roll, 

One starry concave fills the midnight air ; 
Whether wc view the Zodiac or the Pole, 

The constellated suns of space are there. 

XIII. 

As fades a stone's splash in the waves around, 

Though suns may darken at light's starting place ; 

While ages roll, and cycles wheel their bound, 
Light speeds, centrifugal, its onward race. 

XIV. 

Heaven's vast machine defies the optician's art, 

Naught but Omniscience its depths may scan, 
What man may know is but a little part, 

All unrcvcalcd to him the glorious plan. 

XV. 

Author of All ! Almighty, yet unseen, 

Wondrous, surpassing wonder Thou must be, 
Thou veil'st thyself beyond the starry scene, 

The light thou mad'st reveals thy works, not Thee. 

XVI. 

Thy Omnipresence shrouds itself in light, 

Where its bright rays illume the furthest sky ; 

The tiny shadow of Earth's little night 
Hides nothing from thine ever-seeing eye. 

[The shadow of a planet, as of Venus, or of the earth, always exists on the side from the 
sun, and it is our rolling into and out of this shadow which causes day and night to us. It 
obstructs the whole sunlight no more than a raven's wing in the furthest horizon. 

[The same laws of radiation from each point to every other point, the same wonderful 
inter-radiations are true of reflected light. Let us suppose a million of observers on a moun- 
tain side, above them the Heavens, before them a rich and varied landscape. Each observer, 
if he looks upward, receives on the expanse of the nerves of vision a daguerreotype, as it 
were, of the constellations. If he looks forward and around, ho has a changing daguerreotype 
of the landscapes. Tho daguerreotype of the constellations may, it is true, be obscured by the 
daylight of our own star, but it is not the less certainly there. 



128 EXPLANATION OF THE PLATES. 

[Provide each of these million of observers with telescopes of great power; each and all can 
now analyze the celestial area, and fix upon any one of the telescopic stars. Its feeblest light 
is present to all. 

[Each and all in concert can next analyze the landscape, and reduce its vast variety to 
distinct vision. They can each direct their telescope to any given point in the field of vision. 
They can, with an engineer's common portable telescope, see the same man forty miles distant. 
Every point in the larger field of vision of the mountaineer, or smaller field of the citizen, 
bounded by the adjacent buildings, is really a centre of reflected light. The universe may be 
6aid to be filled with the direct radiant light from each of the fixed stars inter-radiating recip- 
rocally with each of the other fixed stars. Through this the reflected light from the moon, 
and from each of the planets, and their satellites inter-radiates freely, as though it were the 
only light in nature. And also the radiation of light from every point of the distant land- 
scape or the adjacent buildings is inter-radiating independently on all sides. 

[The eye is the most delicate and accurate of optical instruments. The telescope lengthens 
the darkened vestibule of its camera-obscura-like chamber, and thus more effectively prevents 
the interference of lateral light. The most complex perspective, the most exquisite foliage of 
the distant landscape, which would be perceptible only by a telescope of a thousand powers, 
are unquestionably pictured with perfect exactness upon the eye of the rudest observer. 

[As the measurement of all radiant phenomena, all forces acting from a centre explain and 
illustrate each other, any contrivance, whether pictorial, arithmetical, algebraic, or poetically 
descriptive, which explains one, aids in explaining all.] 

Our design to aid the student in forming a general concep- 
tion of the great truths which it is the purpose of mathematics 
to unravel, and to identify the three varieties of measurement, 
linear, superficial and massive, and their relation to motion, 
rectilinear, curvilinear and radiant, may now be to some ex- 
tent recognised. The subject is too comprehensive to be 
easily exhausted, but the author feels that he must bring 
this volume to an abrupt conclusion, asking that it be read 
connectedly, with some indulgence for imperfections incident 
to a first edition. 

It is not improbable that some important extension of the 
work is essential to its completeness ; but brevity is the soul 
of wit, and the present volume is submitted with this merit to 
the judgment of the public. The author hopes that fewer 
imperfections may be found in succeeding editions. 



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